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Contents
On local operations that preserve symmetries
and on preserving polyhedrality of maps
Gunnar Brinkmann, Heidi Van den Camp . . . . . . . . . . . . . . . . . . 155
Algebraic degrees of 2-Cayley digraphs over abelian groups
Yongjiang Wu, Jing Yang, Lihua Feng . . . . . . . . . . . . . . . . . . . . 187
A non-associative incidence near-ring with a generalized Möbius function
John Johnson, Max Wakefield . . . . . . . . . . . . . . . . . . . . . . . . 207
Valuations and orderings on the real Weyl algebra
Lara Vukšić . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Regular dessins with moduli fields of the form Q(ζp, p
√
q)
Nicolas Daire, Fumiharu Kato, Yoshiaki Uchino . . . . . . . . . . . . . . . 273
Generalized X-join of graphs and their automorphisms
Javad Bagherian, Hanieh Memarzadeh . . . . . . . . . . . . . . . . . . . . 293
The automorphism group of the zero-divisor digraph
of matrices over an antiring
David Dolžan, Gabriel Verret . . . . . . . . . . . . . . . . . . . . . . . . . 317
Quotients of skew morphisms of cyclic groups
Martin Bachratý . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Finite simple groups on triple systems
Xiaoqin Zhan, Xuan Pang, Suyun Ding . . . . . . . . . . . . . . . . . . . 347
There is a unique crossing-minimal rectilinear drawing of K18
Bernardo M. Ábrego, Silvia Fernández–Merchant, Oswin Aichholzer,
Jesús Leaños, Gelasio Salazar . . . . . . . . . . . . . . . . . . . . . . . . 355
Volume 24, Number 2, Spring/Summer 2024, Pages 155–383
xi

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.01 / 155–186
https://doi.org/10.26493/1855-3974.2749.b64
(Also available at http://amc-journal.eu)
On local operations that preserve symmetries
and on preserving polyhedrality of maps
Gunnar Brinkmann , Heidi Van den Camp *
Ghent University, Krijgslaan 281 S9, Ghent, Belgium
Received 1 December 2021, accepted 8 July 2023, published online 4 September 2023
Abstract
We prove that local operations that preserve all symmetries, as e.g. dual, truncation, me-
dial, or join, as well as local operations that are only guaranteed to preserve all orientation-
preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps.
This generalizes a result by Mohar proving this for the operation dual. We give the proof
based on an abstract characterization of these operations, prove that the operations are well
defined, and also demonstrate the close connection between these operations and Delaney-
Dress symbols.
Keywords: Embedded graph, map, polyhedral embedding, operation, symmetry, tiling.
Math. Subj. Class. (2020): 05C76, 05C10, 05C40, 52B05
1 Introduction
Symmetry-preserving operations on polyhedra have been studied for a very long time. They
were first applied in ancient Greece. Some of the Archimedean solids can be obtained from
Platonic solids by applying the operation which was later called truncation by Kepler. Over
the centuries, polyhedra and specific operations on them have been studied extensively
[3, 11, 12, 18, 22]. However, a general definition of the concept local symmetry-preserving
operation and a systematic way of describing such operations was only presented in 2017
[2]. This description covers a large class of operations on maps, including all well-known
symmetry-preserving operations such as truncation, dual, or those operations known as
achiral Goldberg-Coxeter operations [4, 5]. Goldberg-Coxeter operations were in fact in-
troduced by Caspar and Klug [4] and can be used to construct all fullerenes or certain
viruses with icosahedral symmetry.
*Corresponding author.
E-mail addresses: Gunnar.Brinkmann@ugent.be (Gunnar Brinkmann), heidi.vandencamp@gmail.com
(Heidi Van den Camp)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
156 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
v2
v1 v0
Figure 1: On the left, the barycentric subdivision of a hexagonal face is shown. In the
middle, the lsp-operation truncation is given and on the right the barycentric subdivision
of the result of applying the operation. The blue shaded area shows one chamber of the
original hexagon.
In addition to these local symmetry-preserving operations (lsp-operations), which pre-
serve all the symmetries of a map, there are also operations that are only guaranteed to pre-
serve the orientation-preserving symmetries. Well-known examples of such operations are
snub and gyro [23], or the chiral Goldberg-Coxeter operations. In [2], a general description
of such local orientation-preserving symmetry-preserving operations (lopsp-operations)
was also presented. The very general way of describing lsp- and lopsp-operations in [2] al-
lows to tackle various problems from a more abstract perspective, and also allows to prove
general theorems about the whole class of operations instead of considering each operation
separately. In this paper we will use the new description to prove that all those operations
(e.g. dual, medial, truncation, snub, . . . ) preserve polyhedrality of maps i.e., if an lsp- or
lopsp-operation is applied to a simple 3-connected map of face-width at least 3, then the
result is also simple and 3-connected and it has face-width at least 3.
As the description in [2] was aimed at a broader audience than just mathematicians, the
approach was described in a more intuitive way. In that article an operation is defined as a
triangle ‘cut’ out of a simple periodic 3-connected tiling, and it is applied by gluing copies
of that triangle into the barycentric subdivision of a map. Another way of looking at it is
that the faces of the barycentric subdivision, which are triangles, are further subdivided into
smaller triangles. This is done in a way that the subdivisions of the faces of the barycentric
subdivision are identical or mirror images of each other, or – in case only orientation pre-
serving symmetries must be preserved – in a way that each pair of two triangular faces of
the subdivision that share the same edge as well as the same face of the map is subdivided
in the same way. In the remainder of this text we will give the conditions for these subdivi-
sions that guarantee that the result is the barycentric subdivision of another map – the result
of the operation. An example of an lsp-operation and its application is shown in Figure 1.
In this article, we will give the more direct definition based on Delaney-Dress symbols
that forms the base of this approach and show the connection to the original description.
We will also show that for every lsp-operation there is an equivalent lopsp-operation, i.e. a
lopsp-operation that has the same result as the lsp-operation when applied to a map.
In [2], it is proved that the result of applying an lsp-operation to a polyhedron – that
is: a simple 3-connected map embedded in the plane [17] – is also a polyhedron. In [14]
this result is also announced for all lopsp-operations. We will modify some concepts that
G. Brinkmann et al.: On local operations that preserve symmetries . . . 157
are used in that paper, but due to some serious problems in that paper we will not use the
results given there.
Originally, lsp- as well as lopsp-operations were only defined for simple plane maps
because of their origin in the study of polyhedra. However, there is no mathematical reason
why these definitions should not be applied to maps with multiple edges or loops and
embeddings of higher genus. The question then arises in how far we can extend the theorem
for 3-connected simple plane maps to 3-connected maps of higher genus.
In general, lopsp-operations do not necessarily preserve 3-connectivity for maps that
are not plane. This is obvious for maps with faces of size 1 or 2, but it is also true for simple
maps in general, even if we require the result to be simple. The most striking example of
a local symmetry-preserving operation that can turn 3-connected maps into (even simple)
maps with lower connectivity is dual. In [1] it is proven that for any k ≥ 1, there exist
embeddings of k-connected simple maps M so that the dual M∗ is simple and has a 1-cut.
However, even dual always preserves 3-connectivity in simple maps of face-width at
least three, as proven in [20]. In Definition 4.1 and Definition 4.8 we will define ck-maps
and ck-operations. A map is ck if it is k-connected, it has face-width at least k, and all
of its faces have size at least k. In this paper we will prove the general Theorem 4.9 from
which the following key result is a corollary. The result in [20] for dual is a special case of
this result. The map O(M) is the result of applying the operation O to the map M :
Corollary 1.1. Let k ∈ {1, 2, 3}. If M is a ck-map, and O is a ck-lsp- or ck-lopsp-
operation, then O(M) is also ck.
This theorem is most interesting and relevant for k = 3. This has two reasons. Firstly,
the set of c3-operations contains all well-known and intensely studied operations. Lsp-
operations that are not c3-lsp-operations were not even included in the original definition
of lsp-operations [2]. Secondly, c3-maps, which are in fact simple embedded 3-connected
maps of face-width at least three, have some very interesting properties. These maps are
also known as polyhedral maps or polyhedral embeddings [20]. They can be defined equiv-
alently as simple maps where every facial walk is a simple cycle and any two faces are either
disjoint or their intersection consists of only one vertex or one edge. As the name suggests,
polyhedral maps are a generalisation of polyhedra to surfaces of higher genus. It turns out
that the key property that these operations preserve is not 3-connectivity but polyhedrality.
This property is equivalent to being simple and 3-connected in the plane, but only in the
plane. The main result of this article follows immediately from Corollary 1.1: If M is a
polyhedral map and O is a c3-lsp- or c3-lopsp-operation, then O(M) is also a polyhedral
map (Theorem 4.10).
In Section 2 we give the definitions of the terminology we will use in this text. It starts
with some basic concepts and then the definitions of lsp- and lopsp-operations are given.
There is some freedom in the way that lopsp-operations are applied. However, in Section 3
we will prove that the result of applying a lopsp-operation is independent of the choices
that are made in its application. Section 4 holds the main results of this paper: We prove
a general result that implies that all lopsp-operations preserve polyhedrality of maps. To
show that the definition of lopsp-operations we give is equivalent to the original definition
in [2], we explore the strong connection between lsp- and lopsp-operations and tilings in
Section 5.
158 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
2 Definitions
There are many different, often equivalent, definitions of a map. A short description is
that a map is a cellular decomposition of a surface into vertices (0-cells), edges (1-cells),
and faces (2-cells). Perhaps more intuitively, a map is an embedding of a topological
representation of a graph G onto a surface S. In this text we will only consider 2-cell
embeddings, which means that all the connected components of S \G are homeomorphic
to 2-dimensional disks. We will only consider oriented surfaces. What we will refer to
as map is often called an oriented map in texts where more general maps are also studied.
Maps are often studied from a topological point of view. To make some technical details
easier to describe rigorously, we will use the combinatorial approach that is given below.
This definition is equivalent to the topological ones [16, 21]. We will define a map as a
graph together with a rotation system, which for every vertex imposes a cyclic rotational
order on the edges incident to that vertex.
A graph is a tuple (V,E) where V is the set of vertices and E is the set of edges.
Every edge is incident to two vertices that are not necessarily different. If they are the same
vertex, then that edge is a loop. Though we are mainly interested in graphs without loops
or multiple edges, they will occur in a natural way — e.g. as tools or as the result of an
operation — so that we will in general assume that the underlying graph of a map may have
multiple edges and loops and explicitly restrict the class where necessary.
With every edge of a graph G, we associate two oriented edges, each starting in one
vertex of the edge and ending in the other. In the literature these are also called directed
edges or darts. If e is one oriented edge, then e−1 is the other oriented edge associated
with the same edge of G. For every vertex v of G, a cyclic order is assigned to all oriented
edges starting at v. This way, every oriented edge e has a ‘successor’ σ(e). A map – also
known as embedded graph or graph embedding – is a connected graph together with such a
successor function σ. In a more general context, our maps could be referred to as oriented
maps. As we will not consider unoriented maps we just use the term map. The vertices and
edges of a map are the vertices and edges of the underlying graph. When drawing maps,
the cyclic order around the vertices induced by σ corresponds to the clockwise order of
edges around that vertex in the drawing.
A map is simple if it has no loops and no multiple edges that are incident with the same
2 vertices.
A map is k-connected if it has at least k + 1 vertices and it has no vertex-cut of fewer
than k vertices.
Consider three oriented edges e1, e2, and e3 incident with a vertex v. We say that e2 is
between e1 and e3 if e1, e2, e3 occur in this order in the cyclic order around v, i.e. the cyclic
order of edges around v is of the form (. . . e1 . . . e2 . . . e3 . . .) and not (. . . e3 . . . e2 . . . e1 . . .).
We say that e and σ(e−1) form an angle in the map. A face of a map M is a cyclic
sequence of oriented edges such that every two consecutive edges form an angle. We will
use the term facial walk to refer to the closed walk in M corresponding to this cyclic
sequence of oriented edges. This definition of face corresponds to the topological notion
of a face.
For a map M , with VM , EM , and FM denoting the sets of vertices, edges, and faces of
M respectively, χ(M) = |VM |−|EM |+ |FM | is the Euler characteristic of M . The genus
of M is defined as gen(M) = 2−χ(M)2 . If a map has genus 0, it is called plane. Note that a
plane map is not the same as a planar graph. A planar graph is a graph (not embedded) that
G. Brinkmann et al.: On local operations that preserve symmetries . . . 159
Figure 2: The left figure shows a map and one of its submaps with bold edges.The right
figure shows the internal component of the only face of the submap that has bridges.
can be embedded such that it has genus 0. A plane map is one specific genus 0 embedding
of a graph.
Let M be a map and G′ a subgraph of the underlying graph of M . The map M ′ which
is the graph G′ with the embedding induced by that of M is called a submap of M .
To formalize when a vertex or edge is ‘in’ a face of one of its submaps we will now
define what a bridge for a submap M ′ of M is. There are two kinds of bridges:
• If e ∈ EM \EM ′ is an edge with endpoints v, w ∈ VM ′ , then the submap with vertex
set {v, w} and edge set {e} is a bridge.
• Let C be a component of the submap of M induced by the vertices of M that are not
in M ′, and define E′C = {e ∈ EM | e ∩ VC ̸= ∅} and V ′C = {v ∈ VM | ∃e ∈ E′C :
v ∈ e}. Then the submap with vertex set V ′C and edge set E′C is a bridge.
If a bridge has an edge that is between two edges e and e′ so that e−1 and e′ form an
angle in a face of M ′, then the bridge is in that face. All the vertices and edges of the bridge
are also said to be in the face. The boundary ∂f of a face f is the submap of M consisting
of all the vertices and edges in the facial walk of f . A vertex or edge of M is in the interior
of a face of M ′ if it is in that face and it is not in the boundary.
If a bridge is in more than one face, we say that those faces are bridged. A face that is
not bridged is called simple.
Let f be a simple face of M ′. We will define the internal component of f as follows.
Start with the submap N of M that consists of the boundary of f together with all bridges
in f . Intuitively, we cut along the boundary of f in N in such a way that the facial walk
becomes a simple cycle. More formally, we replace every vertex v of N that appears k > 1
times in the facial walk of f by k pairwise different vertices v1, ..., vk. If both oriented
edges associated with an edge of M ′ appear in the facial walk, this edge is also split into
two different edges between different copies of its vertices. Let (x, v) and (v, y) be the
oriented edges that form the angle in M ′ at the i-th occurrence of v. Then we define the
rotational order (and also the neighbours) of vi to be the same as the rotational order around
v in M , but restricted to the edges between (v, x) and (v, y). Of course some vertices may
be replaced by their copies. The result of this is the internal component IC(f) of f . An
example of an internal component is illustrated in Figure 2. If IC(f) is plane, we call f
internally plane.
An important concept in the definition of lsp- and lopsp-operations is the barycentric
subdivision of a map. It is obtained by subdividing every face into triangular faces, which
we will call chambers. We will also use the barycentric subdivision to define contractible
cycles and face-width in a combinatorial way.
160 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
Figure 3: A face in a map M and the corresponding part of BM . Edges of colour 1 are
dashed and edges of colour 2 are dotted.
The barycentric subdivision BM of a map M is a map that has a unique vertex for every
vertex, for every edge and for every face of M . We always assume that BM comes with
the natural vertex-colouring that assigns colours 0, 1, and 2 to vertices that correspond to
vertices, edges, and faces of M respectively. These colours correspond to their topological
dimension. There are edges between vertices of colour 0 and 1 if the corresponding vertex
and edge are incident. There are edges between vertices of colour 0 or 1 and colour 2 if
the corresponding vertex or edge appears in the boundary of the corresponding face. There
are no edges between vertices of the same colour. For i ∈ {0, 1, 2}, an edge is of colour
i if it is not incident with a vertex of colour i. We will also refer to vertices and edges of
colour i as i-vertices and i-edges. The rotational order of the edges adjacent to a vertex of
colour 2 follows the order of the vertices and edges in the corresponding facial walk of M ,
and similarly for vertices of colour 0 and 1. This is illustrated in Figure 3. Every face of
BM is a triangle. Note that in every figure in this text, colours are represented by colours
in the order rgb, that is: a red is colour 0, green is colour 1, and black is colour 2. The
edges of colour 1 are dashed and the edges of colour 2 are dotted, so that when looking
at the figures printed in black and white it should still be clear which edges have which
colour. With this rotation system, a short calculation of the Euler characteristic shows that
gen(BM ) = gen(M). If x is a face, edge or vertex of M , then to keep notation simple we
will also write x for the corresponding vertex of BM .
Every face of BM is a triangle, with exactly one vertex and one edge of each colour.
We call such a triangle a chamber. Two chambers are adjacent if they share an edge. In the
literature, chambers are also called flags. The flag graph of M is the dual of BM , i.e. it is
the 3-regular graph that has the chambers as its vertices, and there is an edge between two
vertices if their corresponding chambers are adjacent. In some papers flags are defined as
triples (v, e, f) where v, e, and f are respectively a vertex, an edge, and a face such that v
is a vertex of e and v and e are in face f [7, 19]. We cannot use that approach here because
with our general definition of a map there is no 1-to-1 correspondence between chambers
and triples (v, e, f). For example, an edge can have the same face on both sides so that
there are multiple chambers with the same vertices.
Lemma 2.1. A map M , vertex-coloured with colours 0,1, and 2 is the barycentric subdi-
vision of another map if and only if:
(i) Every face of M is a triangle.
G. Brinkmann et al.: On local operations that preserve symmetries . . . 161
(ii) There are no edges between vertices of the same colour.
(iii) Every vertex of colour 1 has degree 4.
Proof. Let VG and EG be the sets of vertices of M with colours 0 and 1 respectively.
Conditions (i) and (ii) imply that every face has exactly one vertex of each colour. It now
follows from (iii) that a vertex e ∈ EG of colour 1 has two neighbours in VG and two
neighbours f and g of colour 2. This induces an incidence relation on the vertex set VG and
the edge set EG that defines a graph G. The rotation system of M induces a rotation system
on G. Let N be the map that consists of G with this rotation system. It is not difficult to
check that M = BN .
The double chamber map DM of a map M is the submap of BM that only contains the
edges of colours 1 and 2. A double chamber of a map M is a face in DM . Every double
chamber has length four: two (in case of no loops different) vertices of colour 0, one of
colour 1, and one of colour 2. Two double chambers are adjacent if they share a 1-edge or
two 2-edges.
In [2], lsp- and lopsp-operations are – following Goldberg [15] – defined in a geometric
way as triangles ‘cut’ out of the barycentric subdivision of a 3-connected tiling of the
plane, such that in case of lsp-operations the sides of the triangle are on symmetry axes
of the tiling. In this article we give purely combinatorial definitions of lsp- and lopsp-
operations, similar to [14] and [13]. The definitions given here are equivalent to those in
[2] when restricted to what we will later call c3-operations. The equivalence can be seen by
applying operations as defined here to some special periodic tiling, but readers who want
to see the equivalence already before starting on the main results of this paper and who
want to have a deeper insight into the relation of operations and periodic tilings encoded
by Delaney-Dress symbols, can find a direct proof without applications of the operations
in Section 5.
Definition 2.2. Let O be a 2-connected plane map with vertex set V , together with a
colouring c : V → {0, 1, 2}. One of the faces is called the outer face. This face contains
three special vertices marked as v0, v1, and v2. We say that a vertex v has colour i if
c(v) = i. This 3-coloured map O is a local symmetry preserving operation, lsp-operation
for short, if the following properties hold:
(1) Every inner face — i.e. every face that is not the outer face — is a triangle.
(2) There are no edges between vertices of the same colour, i.e. the colouring is proper.
(3) For each vertex that is not in the outer face:
c(v) = 1 ⇒ deg(v) = 4
For each vertex v in the outer face, different from v0, v1, and v2:
c(v) = 1 ⇒ deg(v) = 3
and
c(v0), c(v2) ̸= 1
162 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
c(v1) = 1 ⇒ deg(v1) = 2
An example of an lsp-operation is shown in the middle of Figure 1.
Just like for barycentric subdivisions we say that an edge is of colour i if it is not
incident to a vertex of colour i. This is well-defined because of the second property.
Every inner face has exactly one vertex and one edge of each colour. We will refer to
these triangular faces as chambers.
In the original paper [2] only operations that preserve 3-connectivity of polyhedra were
discussed, so the result of the operation also had to have only vertices of degree at least
3. In [13] operations were also discussed that produce maps with 1- or 2-cuts, but the
restriction that vertices in the result should have degree at least 3 was kept. Our definition
of lsp-operations is even more general. With this definition, the result of applying an lsp-
operation may have vertices of degree 1 or 2.
Application of an lsp-operation:
Let O be an lsp-operation and let M be a map. The operation is applied to M by first
replacing for i ∈ {0, 1, 2} the i-edges of BM by copies of the part of the boundary of the
outer face of O between vj and vk with i ̸= j, k. The copy of vj is identified with the j-
vertex and the copy of vk with the k-vertex. Then — depending on the orientation — either
a copy of O or a copy of the mirror image of O — which has the same underlying graph
as O but the rotation system is the inverse of that of O — is glued into every face of the
modified BM . Note that chambers of BM sharing an edge have different orientations. The
boundary vertices are identified with their copies. This results in a 3-coloured triangulation.
An example of the gluing — restricted to a single face — is given in Figure 1. With
Lemma 2.1 and Definition 2.2 it follows that this triangulation is the barycentric subdivision
of a map O(M), the result of applying O to M .
As any symmetry group acts on the chamber system, lsp-operations preserve all the
symmetries of a map. New symmetries can also occur. However, all known examples of
3-connected maps where lsp-operations can increase symmetry are maps of genus at least 1
or they are self-dual. It is an open question whether lsp-operations can increase symmetry
in plane 3-connected maps (polyhedra) that are not self-dual.
There are also interesting operations such as gyro and snub that are only guaranteed to
preserve the orientation-preserving symmetries of maps. These cannot be described by lsp-
operations. In the supplementary material of [2] and in [14], local orientation-preserving
symmetry-preserving operations (lopsp-operations) are defined similarly to lsp-operations.
The most important difference is that here the decoration is glued into double chambers
instead of chambers. As with lsp-operations, we will give a more explicit definition of
lopsp-operations that is not directly based on tilings.
There are some problems that arise in the original definition of lopsp-operations that
do not appear for lsp-operations. With the original definition, it is possible to cut different
patches out of a tiling that describe the same operation and must be shown to have the
same result. That is why we define a lopsp-operation as a plane triangulation, similar to
[14], and not as a quadrangle that we can glue directly into double chambers. Although
this simplifies the definition of a lopsp-operation, the same problem comes back when it is
described how the operation is applied.
Definition 2.3. Let O be a 2-connected plane map with vertex set V , together with a
colouring c : V → {0, 1, 2} and three special vertices marked as v0, v1, and v2. We say
G. Brinkmann et al.: On local operations that preserve symmetries . . . 163
v0
v1
v2
v2
v1
v0,L v0,R
Figure 4: On the left, the lopsp-operation gyro is shown. The thick edges are the edges of
the path P . On the right the corresponding double chamber patch OP is drawn.
that a vertex is of colour i if c(v) = i. The map O is a local orientation-preserving
symmetry-preserving operation, lopsp-operation for short, if the following properties hold:
(1) Every face is a triangle.
(2) There are no edges between vertices of the same colour, i.e. the colouring is proper.
(3) For each vertex v different from v0, v1, and v2:
c(v) = 1 ⇒ deg(v) = 4
and
c(v0), c(v2) ̸= 1
c(v1) = 1 ⇒ deg(v1) = 2
Again we say that an edge has colour i if it is not incident to a vertex of colour i and this
is well-defined because of the second property. Note that the edges incident with a vertex
have two different colours, and as every face is a triangle, these colours appear alternatingly
in the cyclic order around the vertex. The requirement that O is 2-connected is mentioned
in the beginning, but would in fact also follow from the other conditions. Again every face
has exactly one vertex and one edge of each colour and will be referred to as a chamber.
The dual of O will be referred to as the flag structure of O.
Application of a lopsp-operation:
For vertices v, v′ in a path P we write Pv,v′ for the subpath of P from v to v′.
As lopsp-operations are 2-connected, due to Menger’s theorem there are two paths, one
from v0 to v1 and one from v0 to v2 that have only v0 in common. These paths together
form a longer path P from v1 to v2 through v0. As a submap of O, P has a single face. In
this facial walk only v1 and v2 occur once and all other vertices of P occur twice. We say
that such a path P is a cut-path of O. Consider the internal component of the only face of
164 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
v2
v1
v0,L v0,R
Figure 5: On the left the barycentric subdivision of a hexagonal face is shown. In the
middle, a double chamber patch OP of the operation gyro is drawn, and the right image
shows the part of BOP (M) corresponding to the hexagonal face. The blue shaded area
shows one double chamber.
submap P . This is the double chamber patch OP . It can be drawn in the plane, so that the
two copies of P form the boundary of the outer face. Figure 4 shows this for the operation
gyro. The result of the cutting is a 4-gon with corner vertices v1, v2, and two copies of v0,
which we will denote as v0,L and v0,R. The flag structure of OP is the flag structure of O
where the edges corresponding to edges of P are removed.
The lopsp-operation is now applied by first replacing the edges of a double chamber
map DM to form the map DM,P . An edge of colour 2 is replaced by a copy of Pv0,v1 and
an edge of colour 1 is replaced by a copy of Pv0,v2 in a way that for i ∈ {0, 1, 2} a copy of
vi is identified with a vertex of colour i.
Gluing copies of the double chamber patch OP into the faces of DM,P — identify-
ing corresponding vertices in DM,P and the copies of double chamber patches — gives
a coloured map BOP (M). Note that the orientation inside a double chamber fixes how
the different copies of v0 have to be identified. Unlike with lsp-operations, we do not use
mirrored copies of O. Figure 5 gives an example — restricted to one face — of this gluing.
A side of a double chamber is a path in the boundary of the corresponding face of DM,P
that is a copy of the path in OP between v2 and v0,L, between v2 and v0,R, or between v0,L
and v0,R. A side is a 1-side if it is between copies of v0 and v2 and it is a 2-side if it is
between two copies of v0.
Lemma 2.4. Let M be a map and let O be a lopsp-operation with a cut-path P . The
3-coloured map BOP (M) is the barycentric subdivision of a connected map.
Proof. This follows immediately from Lemma 2.1.
As lsp-operations preserve all symmetries of a map, they also preserve the orientation-
preserving symmetries, so one would expect that for every lsp-operation, there is a lopsp-
operation that has the same result when applied to any map. This observation allows to
prove some properties of the result of applying lsp- or lopsp-operations only for lopsp-
operations. The result for lsp-operations can then be deduced from the corresponding
lopsp-operation. Such an equivalent lopsp-operation can be obtained in the following way:
Let O be an lsp-operation, and let c be the boundary of the outer face of O. Let Olopsp
be the map obtained by gluing a mirrored copy of the inner face of c into the outer face,
identifying the vertices on c with their copies. The vertices v0, v1, and v2 of O are also the
vertices v0, v1, and v2 of Olopsp.
G. Brinkmann et al.: On local operations that preserve symmetries . . . 165
Lemma 2.5. If O is an lsp-operation, then Olopsp is a lopsp-operation, and O(M) =
Olopsp(M) for any map M .
Proof. Olopsp is obviously a triangulation of the disc and there are no edges between ver-
tices with the same colour. As Olopsp consists of two copies of O, glued along the boundary
c, we can associate a unique vertex o(x) of O with every vertex x of Olopsp. The degree of
x in Olopsp is given by
deg(x) =
{
deg(o(x)) if o(x) is not in c
2deg(o(x))− 2 if o(x) is in c
.
From the degree restrictions for lsp-operations we can now deduce the degree restric-
tions in the definition of lopsp-operations for Olopsp. It follows that Olopsp is a lopsp-
operation.
Choosing the cut-path in Olopsp that corresponds to the path from v1 to v2 through v0 in
c for the application of Olopsp shows immediately that the results of applying O and Olopsp
are isomorphic: a double chamber is filled in the same way by Olopsp as two adjacent
chambers are filled by O.
3 The path invariance of lopsp-operations
The cut-path chosen to apply an operation is far from unique, so there are many ways
to apply a single lopsp-operation. In this section it is proved that although the ways in
which the operation is applied differ, the result of applying a lopsp-operation to a map is
independent of the chosen path. An essential tool in proving this are chamber flips, which
simulate homotopic deformations.
Definition 3.1. Let P be a directed walk in a barycentric subdivision or lopsp-operation.
For any two different vertices of a chamber C, there are two different simple paths P0, P1
between these vertices in the boundary of C. If for i ∈ {0, 1} path Pi occurs at a certain
position in P , then a chamber flip of C (at this position) is the operation of replacing Pi by
P1−i.
As a first tool we will discuss transformations of one path into another:
Lemma 3.2. Let P, P ′ be two directed paths of the form P = PsR, P ′ = PsR′ from x to
y in a lopsp-operation T , so that R′R−1 is the facial walk of an internally plane face f in
the submap of T consisting of the vertices and edges of P and P ′.
Then there is a sequence of paths P = P0, P1, . . . , Pk = P ′ so that for 1 ≤ i ≤ k
path Pi is obtained from Pi−1 by a chamber flip and every vertex of Pi is in Ps or in the
boundary or the interior of f . As chamber flips can be reversed, the same is true with the
role of P and P ′ interchanged.
Proof. We will prove this by induction on the number |C | with C the set of chambers of T
inside f . If |C | = 1, then R and R′ are the two paths along the boundary of a chamber, so
one can be transformed into the other by one chamber flip and we are done. Now assume
that |C | ≥ 2. We prove that there are at least two chambers in C that have a connected
intersection with ∂f that contains at least one edge: Let Ff be the dual of T restricted to
166 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
C and without edges that correspond to edges in ∂f . If T is the barycentric subdivision
of a map then Ff is part of the flag graph of that map. There are at least two chambers
in C that contain an edge of ∂f . Assume that there is a chamber C such that C ∩ ∂f is
disconnected. This chamber C splits the set C into two parts, i.e. the vertex corresponding
to C is a cut-vertex of Ff . In each component of Ff \ C there is at least one chamber
that shares an edge with ∂f . Let C0 be a chamber that contains an edge of ∂f that has
the largest distance dmax to C along a path in Ff . If this chamber has a disconnected
intersection with ∂f , then its corresponding vertex is a cut-vertex of Ff . This implies
that there is a chamber that shares an edge with ∂f and has a larger distance to C than
dmax, which is in contradiction with the maximality of dmax. Repeating this argument for
the other component of Ff \ C, it follows that in each of the two components there is a
chamber that has a connected intersection with the facial walk ∂f that contains at least one
edge.
Assume that one of these two chambers intersects ∂f in a single edge or in two edges
of P or of P ′. Then we can do a chamber flip to obtain either a path P1 or Pk−1, so that
we can apply induction to P1, P ′ or P, Pk−1 and use that each chamber flip can be undone
by a reverse chamber flip.
If the intersection of neither of the two chambers with ∂f is one or two edges of P
or P ′, then both intersections consist of one edge of P and one edge of P ′. For one of
the chambers, the shared vertex of those edges is the first vertex of R and R′. Applying a
chamber flip replacing the edge of P , we get a path P1 to which we can apply induction.
Lemma 3.3. Let Q,Q′ be two directed paths from x to y in a lopsp-operation, and z a
vertex not contained in either of the paths.
Then there is a sequence of paths Q = Q0, Q1, . . . , Qk = Q′ from x to y so that for
1 ≤ i ≤ k the path Qi is obtained from Qi−1 by a chamber flip and none of the paths
contain z.
Proof. We will prove this by backwards induction on the number n of edges in the begin-
ning of Q that Q and Q′ have in common. Remember that for vertices v, v′ in a path Q we
write Qv,v′ for the subpath of Q from v to v′.
If n = |Q′|, then Q = Q′, so assume that n < |Q′| and that the assumption is true for
n′ > n. Then there is a first vertex a in Q that is incident with an edge that is in Q′ but
not in Q. Let b be the next vertex after a in Q′ that Q′ shares with Q. We will show that Q
can be transformed to Q′x,aQ
′
a,bQb,y in the described way, so that we can apply induction
to transform Q′x,aQ
′
a,bQb,y into Q
′.
Let c be the cycle Qa,b ∪Q′a,b. We call the face of c containing z the exterior. Note that
neither Q′x,a = Qx,a nor Qb,y intersects c in a vertex other than a or b.
There are four possibilities for the position of Qx,a and Qb,y . These are depicted in
Figure 6. If Qx,a or Qb,y are in the interior of c, we use them as part of the face boundary
when applying Lemma 3.2, otherwise we do not. As Lemma 3.2 already allows to consider
paths that start with a common part outside the face, we can choose P, P ′ from Lemma 3.2
in the following way:
Qb,y outside: Choose P = Qx,aQa,b, P ′ = Q′x,aQ
′
a,b.
Qb,y inside: Choose P = Qx,aQa,bQb,y , P ′ = Q′x,aQ
′
a,bQb,y .
G. Brinkmann et al.: On local operations that preserve symmetries . . . 167
Q′a,b
Qx,a
x
a b
y
z
Qb,y
Qa,b
Q′a,b
Qa,x
x
a b
y
z
Qb,y
Qa,b
Q′a,b
Qx,a Qb,y
x
a b
y
z
Qa,b
Q′a,b
Qx,a Qb,y
x
a b y
z
Qa,b
Figure 6: The four different cases in the proof of Lemma 3.3 are shown here. The shaded
area represents the interior.
Note that in case Qx,a is outside c it forms the Ps from Lemma 3.2, otherwise Ps
consists of a single vertex. In each case Lemma 3.2 can be applied to prove that Q can be
transformed to Q′x,aQ
′
a,bQb,y in the described way, and as the beginning of Q
′
x,aQ
′
a,bQb,y
has more than n edges in common with Q′, we can apply reverse induction.
Let M be a map, O a lopsp-operation with cut-path P and OP the corresponding dou-
ble chamber patch. Recall that BOP (M) is obtained by gluing copies of OP into DM .
Therefore every vertex v in BOP (M) is in at least one copy of OP . If v is in more than
one copy, v corresponds to the same vertex of O in each of these copies. Similarly, every
edge or face of BOP (M) also corresponds to exactly one edge or face of O respectively.
This allows us to define a surjective mapping πP , that maps every vertex, edge, and face of
BOP (M) to its corresponding vertex, edge, or face of O.
The mapping πP is not a bijection, but we can define a kind of inverse function π−1P . It
maps a set X of vertices, edges or faces in O to the set of all the vertices, edges or faces in
BOP (M) whose image under πP is in X . If we apply π−1P to a single vertex, edge or face
x of O, we will often write π−1P (x) instead of π
−1
P ({x}). For submaps M ′ of O the image
π−1P (M
′) is a subset of vertices and edges of BOP (M). If these form a connected graph,
we interpret it as a map with the embedding induced by BOP (M).
The definition of BOP (M) depends on P . We will now prove that the result of an
operation is independent of P , so that we can define O(M) for a lopsp-operation O.
Theorem 3.4. Let O be a lopsp-operation and let P and Q be two cut-paths in O. Let M
be a map. Then BOP (M) ∼= BOQ(M).
Proof. The idea of this proof is as follows. We define a submap BOP (M)|Q of BOP (M)
and prove that the underlying graph of this map is isomorphic as a graph to DM,Q. Then we
prove that they are also isomorphic as maps, and that the internal component of each face
of BOP (M)|Q is isomorphic to OQ. It follows that BOP (M) is isomorphic to BOQ(M).
168 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
Let e be an edge of DM , and let j be 1 if e has colour 2 and 2 if e has colour 1. Let
P e be the copy of Pvj ,v0 in BOP (M) that replaced e. By Lemma 3.3 there is a series of
paths Pvj ,v0 = P0, . . . , Pk = Qvj ,v0 from vj to v0 in O, so that the path Pi+1 is obtained
from Pi by a chamber flip of a chamber Ci and none of v0, v1, v2 occur as interior points
of any of the paths. We define a sequence of paths P e = P e0 , . . . , P
e
k in BOP (M) with
πP (P
e
i ) = Pi for 0 ≤ i ≤ k. The path P ei+1 will be obtained from P ei by applying a
chamber flip to a chamber C ∈ π−1P (Ci). The chamber flips in O on the paths Pi replace
subpaths of one or two edges. In case of one edge it is clear that a corresponding chamber
flip can be performed on P ei in BOP (M). In case of two edges, we have to prove that
the two corresponding edges of P ei are also contained in the same chamber. As P
e
i is a
path, the two edges share one of their vertices, say v. By definition of the paths Pi we get
that πP (v) /∈ {v0, v1, v2}. For such a vertex v it is true that if e1, e2, . . . , ek is the cyclic
order of edges around v, then πP (e1), πP (e2), . . . , πP (ek) is the cyclic order of the edges
around the vertex πP (v). If a chamber flip is applied to the edges πP (ej) and πP (ej+1) to
go from Pi to Pi+1, then we can apply a chamber flip to the edges ej and ej+1 to go from
P ei to a new path P
e
i+1. Thus our sequence of paths P
e = P e0 , . . . , P
e
k in BOP (M) with
πP (P
e
i ) = Pi is defined for 0 ≤ i ≤ k and πP (P ek ) = Qvj ,v0 . We denote P ek as Qe. Note
that Qe is isomorphic to Qvj ,v0 , not to Q. Let BOP (M)|Q be the map consisting of all
the vertices and edges of BOP (M) contained in Qe for some edge e. With the rotational
orders induced by BOP (M) we have that BOP (M)|Q is a submap of BOP (M). First we
prove that as (non-embedded) graphs, BOP (M)|Q and DM,Q are isomorphic.
Two paths Qe and Qe
′
can only intersect in their endpoints: Every other vertex v of
Qe and Qe
′
satisfies πP (v) ̸∈ {v0, v1, v2}, which implies that v has only two incident
edges that are mapped to edges in Q by πP . It follows that two paths of the form Qe are
either disjoint — except possibly for their endpoints — or identical. We prove by induction
that P ei and P
e′
i are disjoint (except for their endpoints) for all 0 ≤ i ≤ k and edges e
and e′ in DM . If e and e′ are edges of a different colour this is trivial as at least one of
their endpoints is different. Assume that e and e′ have the same colour. By our previous
argument it suffices to show that their first edge is different. For i = 0 this is clear. Assume
that it is true for i − 1. Let εi and ε′i be the first edges of P ei and P e
′
i respectively. We
can assume that they are both incident with the same vertex x ∈ π−1P ({v0, v1, v2}). The
paths P ei and P
e′
i are obtained from P
e
i−1 and P
e′
i−1 by one chamber flip for each path.
Either εi = εi−1 and ε′i = ε
′
i−1, or the chamber flips replace εi−1 and ε
′
i−1 by both their
previous edges or both their next edges in the rotational order around x. As εi−1 and ε′i−1
are different edges, εi and ε′i are also different edges, which proves our statement.
It follows that BOP (M)|Q and DM,Q are isomorphic as graphs. Next we prove that
they are also isomorphic as maps.
Let m denote the total number of chamber flips necessary to transform first Pv1,v0 to
Qv1,v0 and then Pv2,v0 to Qv2,v0 . With every face (that is: double chamber) D of DM
and 0 ≤ i ≤ m we can now associate a closed walk Wi that consists of the four paths
P e1i , P
e2
i , P
e3
i , P
e4
i in BOP (M) where e1, . . . , e4 are the four edges of D, in the same
order as they appear in D.
Claim: BOP (M)|Q is a submap of BOP (M) that is isomorphic as a map to DM,Q
and the internal component of each face is isomorphic to OQ.
Let C be the set of all chambers in BOP (M), and let n be the number of chambers in
O. We will define functions αi : C → Z (0 ≤ i ≤ m) with the following properties:
G. Brinkmann et al.: On local operations that preserve symmetries . . . 169
C C
C C
αi+1(C) = αi(C)− 1
αi+1(C) = αi(C) + 1
αi+1(C) = αi(C) + 1
αi+1(C) = αi(C)− 1
Figure 7: The evolution of α after chamber flips. The bold paths with arrows are Wi and
Wi+1.
(i) Let C,C ′ in BOP (M) be two adjacent chambers sharing the directed edges e and
e−1, so that C is on the left of e. For e′ ∈ {e, e−1} we define ni(e′) as the number
of times e′ occurs in the cyclic walk Wi. Then αi(C)− αi(C ′) = ni(e)− ni(e−1).
(ii) For every chamber C in O:
∑
C′∈π−1P (C)
αi(C
′) = 1
As a consequence of (ii) we have
∑
C∈C αi(C) = n.
The walk W0 is an internally plane facial walk of DM,P with an internal component
that is isomorphic to OP . We define α0(C) = 1 if C is a chamber on the inside of W0 and
α0(C) = 0 if C is on the outside. As W0 has exactly one copy of each chamber in O inside
we get (ii) for α0. As α0 only differs for neighbouring chambers if they share an edge of
W0, and then in the way described by (i), we also get (i).
For i > 0 we define αi inductively. Let C be the chamber of O to which a chamber flip
is applied when changing Wi−1 to Wi. These chamber flips occur in two places of Wi−1,
and in fact in different directions. Two chambers C−, C+ with πP (C−) = πP (C+) = C
are involved, C− on the left of the cyclic walk and C+ on the right. We now define
αi(C
−) = αi−1(C
−) − 1 and αi(C+) = αi−1(C+) + 1. This is illustrated in Figure 7.
As we once add one and once subtract one for two chambers with the same image under
πP , (ii) is immediate. Property (i) can be checked easily by looking at αi for C−, C+, and
the neighbouring chambers sharing an edge with them.
For i = 0, The function αi describes whether a chamber is inside or outside Wi. For
other i this is not always the case. If Wi self-intersects the intuitive meaning of αi is less
obvious.
For j = 1 or j = 2, the two edges of Wi incident to the j-vertex x of D are always
moved in the same direction by the chamber flips. This implies that {αi(C) | x ∈ C} is
the same set for every 0 ≤ i ≤ m. As {α0(C) | x ∈ C} ⊆ {0, 1} — it can be {1} if
170 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
M has a loop — every chamber that contains x is mapped to 0 or 1 by αm. The degree in
Wm of every vertex that is not in π−1P ({v0, v1, v2}) is two, so we can follow Wm from v1
and from v2 to the copies of v0 to conclude that for each edge of Wm, the two chambers C
and C ′ containing it have αm(C) ∈ {0, 1} and αm(C ′) ∈ {0, 1}. The αm values of two
adjacent chambers can only differ if their shared edge is in Wm, so 0 and 1 are the only
values of αm. Note that this argument only works because every vertex of Wm that is not
in π−1P ({v0, v1, v2}) has degree 2 in Wm. For 0 < i < m this is not necessarily the case,
and for those values of i the mapping αi may have values different from 0 and 1.
Consider the submap Fm of the dual of BOP (M) — i.e. the flag graph of the map N
such that BN = BOP (M) — induced by the chambers C with αm(C) = 1, where edges
in that map corresponding to edges in Wm are removed. By (ii) it follows that for every
chamber CO in O there is exactly one vertex in Fm. For every edge in the flag structure of
OQ there is exactly one edge in Fm, as by (i) adjacent chambers have the same value under
αm if their shared edge is not in Wm. In fact, these are all the edges of Fm: The maximum
degree of a vertex in Fm is 3, as a chamber is adjacent to three others. Let k be the number
of edges in Q. As there are 2k edges in Wm, we get 2·|EFm | =
∑
v∈Fm deg(v) ≤ 3n−2k.
We also have 2|EFm | ≥ 2|EOQ | = 3n− 2k and thus |EFm | = |EOQ |. It follows that the
flag structure of OQ is isomorphic to Fm. As there are no edges in the flag structure of OQ
that correspond to edges of Q, the walk Wm is the facial walk of a face of BOP (M)|Q.
It follows that BOP (M)|Q and DM,Q are isomorphic maps and the internal component of
each face of BOP (M)|Q is isomorphic to OQ.
Definition 3.5. Let O be a lopsp-operation and let M be a map. Choose any cut-path
P in O. The result O(M) of applying O to M is the map with barycentric subdivision
BOP (M).
By Lemma 2.4 and Theorem 3.4, O(M) is well-defined and independent of the chosen
path. We can also define the map π := πP as it is independent of the chosen path.
4 The effect of lsp- and lopsp-operations on polyhedrality
Polyhedral maps are simple maps that are 3-connected and have ‘face-width’ at least three.
The face-width (or representativity) of a map is a measure of ‘local planarity’. Embeddings
of high face-width share certain properties with plane maps. We will define face-width in
a combinatorial way, using barycentric subdivisions. It is not difficult to prove that the
definition given here is equivalent to the definition in e.g. [20].
A cycle in a map M is contractible if – as a submap of M – it has a simple internally
plane face. The face-width of a map M , denoted fw(M), is the minimal length of a non-
contractible cycle in BM , divided by two. If M has no non-contractible cycles, i.e. M is
plane, then we define fw(M) = ∞.
Definition 4.1. For k ≥ 1 we define a map M to be ck if:
• M has no cut-sets with fewer than k vertices
• fw(M) ≥ k
• The size of every face of M is at least k
• The degree of every vertex of M is at least k
G. Brinkmann et al.: On local operations that preserve symmetries . . . 171
The condition that neither cuts with fewer than k vertices nor vertices with degree
smaller than k may be present instead of just requiring the map to be k-connected is chosen
in order to deal with small boundary cases. For example, a cycle is 2-connected, but its
dual is a map with only two vertices so it is not 2-connected. Both a cycle and its dual are
c2.
A polyhedral map is a simple, 3-connected map that has face-width at least three.
Lemma 4.2. A map is c3 if and only if it is polyhedral.
Proof. It suffices to prove that every c3-map is simple and has at least four vertices. The
rest of the statement is trivial when the definitions are written out. Let M be a c3-map.
Facial loops and facial 2-cycles are excluded by the restrictions on face sizes and non-facial
loops and non-facial 2-cycles imply either smaller cuts or a smaller face-width. Therefore
M is simple. It has at least 4 vertices as it has minimum degree at least 3 and it is simple.
The reason why the term c3 is used instead of polyhedral in this article is that many
results are proven for ck maps for general k ∈ {1, 2, 3}.
The following lemma characterises c2- and c3-maps by a condition based on the cham-
ber system. A 4-cycle in a barycentric subdivision is called trivial if it has a face that has
no vertex or only a single colour-1 vertex in its interior.
Lemma 4.3. Let M be a map.
(i) M is c2 if and only if BM has no cycles of length 2.
(ii) M is c3 if and only if M is c2, and BM has no nontrivial cycles of length 4.
Proof. (i): Let M be a map and assume that M is not c2. There are four possible reasons
for not being c2: the existence of a cutvertex, the existence of a facial loop (a face of size 1),
the existence of a vertex of degree 1, or the existence of a non-contractible 2-cycle in BM .
The last three immediately imply the existence of a 2-cycle in BM , so assume that M has a
cutvertex v. If there is a loop in M , then there is a 2-cycle in BM , so assume that M has no
loops. Then vertex v has neighbours x and y in different components such that y follows x
in the rotational order around v. The facial walk (x, v), (v, y), (y, w1), . . . , (wk, x) of the
face f containing this angle must also contain v as one of the wi as otherwise part of the
facial walk would be a path from x to y in M \ {v}. This implies that in the barycentric
subdivision there are 2 edges between v and the vertex corresponding to f — a 2-cycle.
Conversely, assume that there is a 2-cycle c in BM . If there is a 0- or 2-vertex of degree
two in BM then there is a vertex of degree 1 or a face of size 1 in M , so we can assume
that every vertex of BM has degree at least four. We can also assume that every cycle of
length 2 in BM is contractible, as otherwise fw(M) = 1 and we are done. This implies
that every 2-cycle has two well-defined sides.
Assume w.l.o.g. that c is innermost, that is: it contains no 2-cycle in its simple, plane
face fc. Let v and w be the vertices of c. Assume that v has colour 1. Then the two neigh-
bours of v that are not w are on different sides of c. Every face has only three edges and
there are no vertices of degree 2, so there are two edges between each of these neighbours
of v and w. This is not possible as c is innermost. It follows that the two vertices of c have
colours 0 and 2 respectively. There is at least one vertex ef of colour 1 in the interior of fc.
This vertex ef has degree 4. If there are no 0-vertices in the interior of fc then there must
172 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
u
v
f1
f2
x
y
Figure 8: This figure clarifies the proof of (ii) of Lemma 4.3. The blue vertices are all in
the same component of M \ {x, y}, and the black vertices are not in that component.
be two edges between ef and the 0-vertex of c. This is a contradiction with the assumption
that c is innermost. It follows that fc has a 0-vertex in its interior. As every 2-cycle has two
well-defined sides, there is an innermost 2-cycle in the other face gc of c. Using the same
arguments as for fc on that cycle we get that gc also has a 0-vertex in its interior. Every
path in BM between the 0-vertices in the two faces using only 2-edges must pass through
the 0-vertex of c. It follows that this vertex is a cutvertex of M , so that M is not c2.
(ii): Let M be a map and assume that M is not c3. If it is also not c2 we are done, so
assume that M is c2. There are four possible reasons for not being c3: the existence of a
cutset {x, y} of size two, the existence of a face of size two, the existence of a vertex with
degree 2, or the existence of a non-contractible 4-cycle in BM . Again the last three, as well
as double edges forming a non-facial 2-cycle in M , immediately give a nontrivial 4-cycle
in BM , so assume that there are no double edges or loops in M , but there is a 2-cut {x, y}.
Both x and y have neighbours in different components. Let u ̸= y be a neighbour of
x, so that the previous vertex in the rotational order around x is not in the same component
of M \ {x, y} as u. Let v be the last vertex, as seen from u, in the rotational order around
x such that v and all vertices in the rotational order between u and v are in the same
component as u, as shown in Figure 8. Note that u = v is possible. If the edges (x, u) and
(v, x) would belong to the same face then there would be a colour-1 cycle of length 2 in
BM , so they are in different faces f1 and f2 of M . Both f1 and f2 must also contain y as
otherwise the next, resp. previous neighbour of x would belong to the same component as
u and v. The cycle x, f1, y, f2 is a nontrivial cycle of length 4 in BM .
Conversely, assume that M is c3 and that M is not c2 or BM has a nontrivial cycle
of length 4. As M is c3 it is also c2, so BM has a nontrivial cycle c of length 4. Note
that there are no double edges in BM as M is c2, and M is simple and 3-connected by
Lemma 4.2.
The cycle c is contractible because fw(M) ≥ 3. It therefore has two well-defined
sides. Assume first that c has no vertices of colour 0 on one side. Then there must be a
2-vertex on that side, as there cannot only be vertices of colour 1. This 2-vertex can have
degree at most 4 as it can be adjacent to at most two 0-vertices in c and there are no double
edges in BM . This implies a facial 2-cycle in M , a contradiction. It follows that there is
at least one 0-vertex on each side of c. Every colour-2 path between 0-vertices on different
sides passes through c. This implies that the vertices and edges of M corresponding to
vertices of c form a cut of M . Ignoring edges if one of their incident vertices is also in c,
G. Brinkmann et al.: On local operations that preserve symmetries . . . 173
this cut consists of 2 vertices, a vertex and an edge or two edges. For each of the edges we
can choose one of its incident vertices such that we find a cut-set consisting of 2 vertices,
which is a contradiction with the 3-connectivity of M .
Lemma 4.3 is very useful to determine whether a map is c2 or c3. It will often be used in
the following lemmas and theorems. The main theorem of this last section is Theorem 4.9,
which shows the equivalence of different definitions of ck-lopsp-operations and states that
when applying ck-lopsp-operations with k ∈ {1, 2, 3} to certain maps, the result is ck. The
most difficult part of its proof is captured in Theorem 4.5 for c2-maps and Theorem 4.7 for
c3-maps.
Lemma 4.4. Let O be a lopsp-operation with a cut-path P of minimal length.
(i) If the vertices of an edge e in O are both in Pv0,vi for an i ∈ {1, 2}, then e or an
edge with the same vertices as e is also in Pv0,vi .
(ii) If the vertices of an edge in OP are in different copies of Pv0,v1 , then there is a
nontrivial 4-cycle in two copies of OP sharing their copies of Pv0,v1 .
Proof. (i): This follows immediately from the minimality of the length of P .
(ii): If Pv0,v1 is v0 = t1, . . . , tk = v1 and we denote one copy with t1, . . . , tk and the other
with t′1, . . . , t
′
k, then — again due to minimality and as O has no loops — such an edge
connects w.l.o.g. ti with t′i+1 for some 1 ≤ i < (k − 1). Considering two copies of OP
sharing the copies of Pv0,v1 , this gives a 4-cycle c = ti, t
′
i+1, t
′
i, ti+1 with v1 in the interior.
If c was trivial, then v1 would be a 1-vertex adjacent to all 4 vertices on c — also ti and t′i
— which contradicts the minimality of P .
Theorem 4.5. Let M be a c2-map and let O be a lopsp-operation. Then O(M) is c2 if
and only if for each cut path P in O we have that there is no 2-cycle in OP .
Note that we must consider 2-cycles in OP and not in O. It is possible that there are 2-
cycles in O that do not induce 2-cycles in OP for any cut-path. For example, the operation
gyro, shown in Figure 4, has several 2-cycles but none in OP for any cut-path P .
Proof. If there is a 2-cycle in OP for a cut-path P then each copy of OP inserted into
DM,P contains a copy of this 2-cycle. Lemma 4.3 now implies that O(M) is not c2.
Conversely, assume that O(M) is not c2. Then there is a 2-cycle c in BO(M). Let
x and y be the vertices of c and let e1 and e2 be its edges. Let P be a cut-path in O
of minimal length. Note that by Lemma 4.3 applied to M , every double chamber has two
different 0-vertices and therefore the boundary of each double chamber is simple. It follows
that if there exists a double chamber that contains both edges of c in its interior or on its
boundary, then c induces a cycle in a copy of OP and we are done. Assume that e1 and e2
are in different double chambers D1 and D2 respectively.
Both D1 and D2 contain x and y, so those vertices are on the boundary of both double
chambers. Assume first that x and y are not both on copies of Pv0,v1 or both on copies of
Pv0,v2 . Then D1 and D2 share their 1-vertex and their 2-vertex which implies a 2-cycle in
BM , a contradiction. It follows that x and y are both on copies of Pv0,v1 or both on copies
of Pv0,v2 . If they are in the same copy of Pv0,v1 or Pv0,v2 , then by Lemma 4.4(i) an edge
e0 with the same vertices is also in that copy of Pv0,v1 or Pv0,v2 , so that OP contains a
2-cycle.
174 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
x y
e1
D1
D2
e2 e′1
Figure 9: This figure clarifies a step in the proof of Theorem 4.5. It shows that a 2-cycle
in BO(M) with its vertices on different copies of the same Pv0,vi cannot exist. The middle
vertex can be the 1- or the 2-vertex of the double chambers. In either case, the left- and
rightmost vertices are the 0-vertices.
t′0=t0
t1t
′
1
tit
′
i
tjt
′
j
tkt
′
k
tst
′
s
Figure 10: The double chamber patch in the proof of Lemma 4.6. There can be no edge
{t′l, tm} with l < j and m > i.
The last possibility is that x and y are in different copies of Pv0,v1 or in different copies
of Pv0,v2 . As O does not contain loops we have π(x) ̸= π(y). Applying Jordan’s curve the-
orem to c we get that the edge e′1 in D2 with π(e1) = π(e
′
1) cannot exist — a contradiction
(see Figure 9).
Lemma 4.6. Let M be a c3-map and let O be a lopsp-operation with a cut-path P of min-
imal length. Let f be a 2-vertex of DM , and consider the submap Sf of BO(M) consisting
of all the edges and vertices in the double chambers with 2-vertex f . If there is a nontrivial
4-cycle c in Sf , then there is a 2-cycle in OP or c is contained in either only one of these
double chambers, or in two adjacent double chambers.
Proof. As M is c3, the submap of all vertices and edges of O(M) belonging to one of
the double chambers containing f is plane. The map formed by the vertices and edges
on the 2-sides of the double chambers in Sf is a simple cycle, that we consider to be the
boundary of the outer face of the map Sf . There are at least three double chambers in Sf .
If c contains only edges on edges of DM , then c is the boundary of one double chamber, so
assume that c contains at least one edge in the interior of a double chamber.
The boundary ∂D of every double chamber D in Sf is a cycle. As Sf is plane, it
follows with the Jordan curve theorem that c must cross ∂D an even number of times. By
cross we mean that there is a subpath of c whose first and last vertex are on different sides
of c, and whose other vertices are all in ∂D. Let D be a double chamber in Sf that contains
G. Brinkmann et al.: On local operations that preserve symmetries . . . 175
an edge of c in its interior. If c crosses ∂D 0 times, then c is contained in one copy of OP
and we are done. If c crosses ∂D 4 times then there must be an edge outside of D that has
both its vertices on ∂D. In this case Lemma 4.4 implies that there is a 2-cycle in OP and
we are done. We can therefore assume that c crosses ∂D exactly twice. Note that every
crossing is on a 1-side. Assume that the vertices of these crossings are on the same 1-side
of D. If there would be only one edge of c in D this again leads to a 2-cycle in OP with
Lemma 4.4. If there is no crossing in f it is clear that a chamber D′ adjacent to D also
has two crossings with c on the same 1-side. If f is in c that follows from the fact that f
is on every 1-side and there must be at least two edges of c in a double chamber that has 2
crossings with c on the same 1-side. It follows that c is completely contained in D and D′,
i.e. in two adjacent double chambers.
We can now assume that c has two crossings with ∂D that are on different 1-sides and
not in f . As c crosses into the double chambers adjacent to D those must also have two
crossings on different 1-sides. Repeating this argument we get that every double chamber
in Sf has two crossings with c on different 1-sides.
It follows that every double chamber in Sf contains a subpath of c connecting vertices
different from f on their two 1-sides. As there are at least three double chambers in Sf
and there must be at least one edge of c in each one, there are exactly three or four double
chambers in Sf . In each case there are at least two adjacent double chambers that contain
only one edge of c. Let e1 and e2 be the only edges of c in two adjacent double chambers.
As the vertices of e1 and e2 are on different 1-sides of OP the edges e1 and e2 are not in
P . Therefore the edges π(e1) and π(e2) induce unique edges, that we will also denote with
π(e1) and π(e2), in OP . If the vertices of Pv2,v0 in O are — in this order — t0, t1, . . . , ts,
then we will denote the vertices on the different 1-sides of OP with t0, t1, . . . , ts, resp.
t′0, t
′
1, . . . , t
′
s. We have π(e1) = {ti, t′j} and π(e2) = {tj , t′k} with w.l.o.g. 0 < i < j ≤ s.
Note that i ̸= j as there are no loops in O. Due to the Jordan curve theorem applied to the
cycle t0 = t′0, t
′
1, . . . , t
′
j , ti, ti−1, . . . , t0 there is no edge {tm, t′l} in OP with m > i and
l < j. As j > i and π(e2) = {tj , t′k} is in OP , it follows that k > j. This situation is
shown in Figure 10.
If there are four double chambers in Sf we can repeat this argument on every pair
of adjacent double chambers. With x1, x2, x3, x4 the vertices of c in cyclic order and
π(xa) = tia we get that ia < ia+1 for all 1 ≤ a ≤ 3 and i4 < i1, so that by transitivity
i1 < i1, a contradiction. If there are only three double chambers in Sf , then there must be
two edges e3 and e4 of c in the same double chamber. The edges π(e3) and π(e4) form a
path from t′i to tk. Such a path would have to cross both the edges π(e1) and π(e2), which
is only possible if the path has at least three edges — it must contain ti or t′j and tj or t
′
k—
which is a contradiction.
Theorem 4.7. Let M be a c3-map and let O be a lopsp-operation. Then O(M) is c3 if
and only if it is c2 and for each cut-path P in O we have that there is no nontrivial 4-cycle
in a patch of two adjacent copies of OP sharing one of their sides.
Proof. The implication that O(M) is not c3 if there is a 2- or nontrivial 4-cycle for some
cut-path P is obvious, as corresponding pairs of two adjacent copies of OP in BO(M)
would contain such cycles.
For the other implication we assume that O(M) is not c3 but it is c2, and that there is
no nontrivial 4-cycle in a patch of two adjacent copies of OP for a cut-path P . We will
come to a contradiction by constructing such a 4-cycle. Let P be a cut-path in O of minimal
176 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
length. We will refer to the copies of v2 or v0 in the double chamber patches as the corners
of the double chamber patches. By Lemma 4.3 there is a nontrivial 4-cycle c in BO(M).
Let X be a set of double chambers in DM of minimal size, so that the union of all double
chamber patches for double chambers in X contains c. For simplicity we will also refer to
the set of those double chamber patches as X . The fact that c has four edges implies that
1 ≤ |X| ≤ 4. If |X| = 1 then c can be thought of as a 4-cycle in OP , which we assumed
does not exist, so |X| > 1. We make the following observations:
(i) Every double chamber in X shares at least two of its corners with other elements of
X: As |X| > 1 the cycle c ‘enters’ and ‘leaves’ any double chamber D ∈ X in two
different vertices. Both of these vertices must be in the intersection of D with the
other double chambers of X .
(ii) If two double chambers share two corners they also share the side containing those
two corners. This follows immediately from the fact that there are no double edges
in M and BM .
(iii) If the intersection of a double chamber D ∈ X with the other double chambers of X
is exactly one side, then there are at least two edges of c in D: Assume that there is
only one edge e of c in D, and let D′ be the double chamber of X sharing the side
with D. Both vertices of e are on the side shared by D and D′, but e itself is not, as
otherwise D could be removed from X and X would not be minimal. If the vertices
of e are on the same edge of DM , then Lemma 4.4 implies that there is a 2-cycle in
OP , a contradiction with Theorem 4.5. If the vertices of e are on different copies of
Pv0,v1 then Lemma 4.4 implies a nontrivial 4-cycle in two adjacent copies of OP , a
contradiction.
It follows from (i) and (ii) that if |X| = 2 then c is a 4-cycle in two adjacent copies of
OP , so |X| > 2. We will now prove that there is a cycle of length 4 in DM that contains at
least one edge of each double chamber in X . We call such a cycle a saturating 4-cycle.
Assume first that every double chamber in X shares the same 2-vertex. With (i), (ii),
and (iii), it follows easily that X consists of all the three or four double chambers corre-
sponding to one face of M . Lemma 4.6 now implies that O(M) is not c2 or that there is a
4-cycle in two adjacent copies of OP . Both are contradictions.
Now assume that there is a 2-vertex f of DM such that there is only one edge e of c
in the union of all the double chamber patches of X with 2-vertex f . The edge e has both
its vertices on the same 2-side. As at least one of its vertices is not a corner, both double
chambers sharing that 2-side are in X . This is a contradiction with (iii).
It follows that there are two 2-vertices f and g in DM such that the unions Xf and Xg
of double chambers patches in X with 2-vertex f and g respectively each contain exactly
two edges of c. With (i), (ii), and (iii) it follows that there are 0-vertices v and w of double
chambers in Xf and Xg such that v, f, w, g is a saturating 4-cycle.
As M is c3, Lemma 4.3 implies that the saturating 4-cycle is the boundary of a double
chamber, or two double chambers sharing the 1-vertex.
If e is the 1-vertex in these one or two double chambers, then c is contained in the set Ne
consisting of all six double chambers that share a side with a double chamber containing e.
We say that the two double chambers with 1-vertex e are the central double chambers, and
the four other double chambers in Ne are the extremal double chambers. The three possible
configurations of Ne with respect to shared sides are shown in Figure 11. Using the fact
G. Brinkmann et al.: On local operations that preserve symmetries . . . 177
e
g
f
e
g
f
e
g
f
Figure 11: The three possible configurations of the six double chambers in the set Ne are
shown here. Different vertices in the drawing represent different vertices of DM .
that there are no nontrivial 4-cycles in BM it can easily be verified that no two vertices in
Figure 11 represent the same vertex of DM .
We already proved that |X| ≥ 3 and that there are two edges of c in Xf and two in
Xg . Therefore we can assume w.l.o.g. that Xf consists of two double chambers. It follows
from (iii) that these two double chambers are extremal double chambers. The two vertices
of the edge e are in c. If the third vertex of c in Xf is f , then with Lemma 4.4 and the fact
that there are no 2-cycles in OP , it follows that the edges of c in Xf are 1-edges of DM .
In that case we can replace the two extremal double chambers in X by the central double
chamber so X was not of minimal size, a contradiction. If the third vertex x of c in Xf
is not f , then the extremal double chambers share a side and x is on this side. Let e1 and
e2 be the edges of c in these double chambers. The edge e1 corresponds to an edge in OP
from w.l.o.g. v0,R to an internal vertex of P(v0,L),v2 and e2 corresponds to an edge from
v0,L to P(v0,R),v2 . This would imply crossing edges in the plane map OP , a contradiction.
It follows that Xf and Xg each consist of only one double chamber, which by (i) must
be the central double chamber. Then c is a nontrivial 4-cycle in a patch of two adjacent
copies of OP , a contradiction.
For a lopsp-operation O, let TO be the tiling of the Euclidean plane obtained by apply-
ing O to the regular hexagonal tiling of the plane. We say TO is the associated tiling of O.
With the definition for lopsp-operations from [2] this is the tiling from which O is defined.
In Section 5 we will further explore the fundamental connection between lopsp-operations
and tilings.
We will use the connectivity of the associated tiling of a lopsp-operation to define when
an operation is ck. With Theorem 4.5 and Theorem 4.7 we will then prove an equivalent
characterisation in Theorem 4.9 which does not depend on the associated tiling.
Definition 4.8. For k ∈ {1, 2, 3} a lopsp-operation O is ck if the associated tiling TO
is k-connected and all faces have size at least k. An lsp-operation is ck if the equivalent
lopsp-operation Olopsp is ck.
For a ck-lopsp-operation O and a map M with minimum face size at least k and min-
imum degree at least k that is not necessarily ck, the map O(M) also has minimum face
size at least k and minimum degree at least k. For the vertices of BO(M) of colour 0 or 2
178 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
that are not 0-vertices or 2-vertices of DM , the fact that they have degree at least 2k follows
from O being a ck-lopsp-operation, as these degrees also occur in the tiling TO. For the
others it follows from the degrees of 0-vertices and 2-vertices in DM . In case M is also
ck, we have a stronger result:
Theorem 4.9. The following statements are equivalent for k ∈ {1, 2, 3} and a lopsp-
operation O:
(a) O is a ck-lopsp-operation
(b) For all ck-maps M , O(M) is ck.
(c) There exists a ck-map M such that O(M) is ck.
Proof. (a) ⇒ (b)
For k = 1 this is trivial. Assume that there is a c2-map M such that O(M) is not c2.
By Theorem 4.5 there is a 2-cycle in OP for some cut-path P in O. This cycle induces a
cycle in BTO , which implies a 1-cut or a face of size 1 in the tiling TO. It follows that O is
not a c2-operation — a contradiction. Similarly, if there is a c3-map M such that O(M) is
not c3, we find a 2-cut in TO using the cycle from Theorem 4.7.
(b) ⇒ (a) For k = 1 this is trivial.
The tiling TO is obtained by inserting copies of OP (for some P ) into the double cham-
ber map of the hexagonal tiling. Let us call the map formed by the subdivided 2-edges of
the double chamber map of the hexagonal tiling the hexagonal skeleton.
Assume now that O is not a c2-lopsp operation. Then there is a face f of size 1 or a
1-cut {x} in TO. In case of a 1-cut, at least one of the components of TO \ {x}, say C0,
is finite. Let C denote a finite submap of the hexagonal skeleton that contains f , resp. C0
together with the cut vertex x. Using Goldberg-Coxeter operations (see [2] or [4]) with
sufficiently large parameters to construct large icosahedral fullerenes, we get a fullerene
F , that is c3 and contains an isomorphic copy of C with the paths between the 0-vertices
replaced by edges. Applying O to that fullerene, we get a submap S of O(F ) that has a
face f ′ of size 1 or that is isomorphic to C0 and where all vertices corresponding to vertices
of C0 have — except for the vertex x′ corresponding to x — only neighbours in S. So f ′
or the vertex x′, which is a cut-vertex of O(F ), are contradictions to the assumption.
The case k = 3 is completely analogous, with also a 2-face and a 2-cut in the argument.
(b) ⇒ (c) Trivial.
(c) ⇒ (b) Note that the conditions in Theorems 4.5 and 4.7 are — except for M being
ck — independent of M , as OP and the union of two copies of OP sharing a side are the
same for all these M . This implies that if O(M) is ck for some ck-map M , then O(N) is
ck for any ck-map N .
One of the main results of this paper, Theorem 4.10, now follows from Theorem 4.9
and Lemma 2.5.
Theorem 4.10. If M is a polyhedral map and O is a c3-lsp- or c3-lopsp-operation, then
O(M) is also a polyhedral map.
G. Brinkmann et al.: On local operations that preserve symmetries . . . 179
σ1
v2
v1 v0
C1
C2
C3
v2
σ0
σ0
σ0
σ2 σ1
m01 m12
C1 12 3
C2 12 3
C3 3 3
Figure 12: The operation truncation and the Delaney-Dress symbol encoding a tiling from
which the operation can be obtained when the original definition is applied.
5 Connection to tilings
In a series of papers [8, 9, 10], Andreas Dress (in later papers together with coauthors)
developed a finite symbol encoding the topology as well as the symmetry of periodic tilings.
He attributed the idea to Matthew Delaney and called these symbols Delaney symbols. In
later papers by other authors, these symbols are called Delaney-Dress symbols. In [6] and
[10] Delaney-Dress symbols of periodic tilings of the Euclidean plane and the hyperbolic
plane are characterized.
In this section we show that there is a very fundamental connection between l(op)sp-
operations and Delaney-Dress symbols and therefore to tilings. Recall that we defined the
associated tiling TO of a lopsp-operation O as the tiling that is the result of applying O to
the hexagonal tiling of the plane, i.e. the tiling with Schläfli symbol {6, 3}. We will find the
same tiling in a different way using Delaney-Dress symbols, and we will see that from a
mathematical point of view the choice of the tiling {6, 3} is quite arbitrary. The hexagonal
tiling was chosen because it was also used in the original definition of lsp-operations in
[2], where in turn it was chosen as a tribute to a paper by Goldberg [15]. By proving
this connection it follows that our abstract combinatorial definitions are equivalent – in the
3-connected case – to the definitions of lsp- and lopsp-operations in [2].
As a topological definition of tilings falls outside the scope of this article, we will
directly start with the combinatorial characterization described in [6, 9, 10]. We will sketch
the connection to tilings, but for a detailed description we refer the reader to [6] or [10].
Theorem 5.1 (A.W.M. Dress [9]). Let D be a set together with an action (from the right) of
the Coxeter group Σ = ⟨σ0, σ1, σ2 | σ2i = 1⟩ on D , and for (i, j) ∈ {(0, 1), (0, 2), (1, 2)}
let mij : D → N be maps with m02(C) = 2 for all C ∈ D . The tuple (D ,m01,m02,m12)
is the Delaney-Dress symbol of a tiling of the Euclidean plane if and only if the following
properties hold:
(1) D has finitely many elements
(2) Σ acts transitively on D
180 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
(3) For i, j ∈ {0, 1, 2}, i < j, mij is constant on ⟨σi, σj⟩-orbits and C(σiσj)mij(C) =
C for all C ∈ D
(4) We have
C (D ,m01,m02,m12) =
∑
C∈D
(
1
m01(C)
+
1
m12(C)
− 1
m02(C)
)
= 0
Such Delaney-Dress symbols encode the combinatorial structure of periodic tilings of
the Euclidean plane, together with a symmetry group acting on the tiling. If C (D ,m01,
m02,m12) ̸= 0, the tuple can also be a Delaney-Dress symbol, but then it encodes a
periodic tiling of the hyperbolic plane (C < 0) or — in case additional divisibility rules
are fulfilled — the sphere (C > 0) [6]. The elements of D are the orbits of chambers of
the tiling under the symmetry group. An element C ∈ D with Cσi = C represents an
orbit of chambers with mirror symmetries of the tiling stabilizing the edges of colour i.
If there are no C ∈ D with Cσi = C, the symmetry group contains no pure reflections,
but maybe sliding reflections. If there are no odd cycles, that is Cσi1 . . . σik ̸= C for odd
k, all symmetries are orientation preserving. The maps m01 and m12 give information
about the symmetry group of the tiling. Let {i, j, k} = {0, 1, 2}, i < j and for C ∈ D
let rij(C) = min{r | C(σiσj)r = C}. Note that rij is constant on ⟨σi, σj⟩-orbits. If a
⟨σi, σj⟩-orbit C⟨σi,σj⟩ contains no C ′ with C ′σi = C ′ or C ′σj = C ′, then the vertices of
colour k of the corresponding chambers in the tiling are centers of an fr-fold rotation with
fr = mij(C)/rij(C). If an orbit C⟨σi,σj⟩ contains a C ′ with C ′σi = C ′ or C ′σj = C ′,
then with fm = 2mij(C)/rij(C) for fm > 1 the vertices of colour k of the chambers in
orbit C are intersections of mirror axes with an angle of 360/fm degrees.
We will now associate a tuple (DO,m01,m02,m12) with an lsp- or lopsp-operation O
and prove that it is a Delaney-Dress symbol. In fact, it will be a Delaney-Dress symbol of
the tiling O(T ) where T is the tiling with Schläfli symbol {6, 3}, i.e. the hexagonal tiling
of the plane where every vertex has degree 3 and every face has 6 edges. Due to the relation
between Delaney-Dress symbols and tilings as described in [6] and [10], this also shows
the equivalence of the combinatorial definitions of lsp- and lopsp-operations defined here
and the geometric ones given in [2]. There a l(op)sp-operation is described as a ‘triangle’
cut out of a tiling in such a way that certain conditions on the symmetry are satisfied.
One could replace the values 3 and 6 we will use for defining the mappings mij by, for
example, 4 and 4, and Theorem 5.3 would still be true. It would however be the Delaney-
Dress symbol of the tiling that can be obtained by applying O to the square tiling of the
plane, which is 4-regular and every face has 4 edges. By using other numbers, other tilings
— even spherical or hyperbolic ones — could be used as source tilings. All of those tilings
can be used to define l(op)sp-operations in the geometric way that was described in [2] for
the hexagonal tiling.
Let O be an lsp-operation and let DO be the set of chambers of O. We define the action
of Σ on DO by letting Cσi = C ′ if C and C ′ share their i-edge, and Cσi = C if the i-edge
of C is in the outer face of O. For (i, j) ∈ {(0, 1), (0, 2), (1, 2)}, let vij(C) be the vertex
of chamber C that is not of colour i or j. We get:
G. Brinkmann et al.: On local operations that preserve symmetries . . . 181
rij(C) = min {r | C(σiσj)r = C} =

∣∣C⟨σi,σj⟩∣∣
2
=
deg(vij(C))
2
if vij(C) is
not in the
outer face∣∣C⟨σi,σj⟩∣∣ = deg(vij(C))− 1 if vij(C) is in
the outer face
To find the Delaney-Dress symbol of the tiling obtained by applying O to {6, 3} we
define mij : DO → N as follows:
mij(C) =

rij(C) · 2 if vij(C) = v1
rij(C) · 3 if vij(C) = v0
rij(C) · 6 if vij(C) = v2
rij(C) if vij(C) /∈ {v0, v1, v2}
Note that the requirements for the vertex degrees in an lsp-operation imply that for all
C ∈ DO, the value m02(C) is 2.
We define D(O) = (DO,m01,m02,m12) and call it the Delaney-Dress symbol cor-
responding to the lsp-operation O. This correspondence is illustrated for the operation
truncation in Figure 12. Theorem 5.2 states that it is in fact a Delaney-Dress symbol of a
tiling of the Euclidean plane.
By our previous remarks there is a 2-fold rotation around each copy of v1 in that tiling,
a 3-fold rotation around each copy of v0, and a 6-fold rotation around each copy of v2.
There are also intersections of mirror axes with 90◦, 60◦, and 30◦ angles at v1, v0, and
v2 respectively. This is the symmetry we expect when applying an lsp-operation to tiling
{6, 3}. This is also the symmetry that is required to define an lsp-operation from a tiling
with the geometric definition.
Theorem 5.2. If O is an lsp-operation, then D(O) = (DO,m01,m02,m12) is the Delaney-
Dress symbol of a tiling of the Euclidean plane.
Proof. We have to prove the properties in Theorem 5.1. The first two properties are obvi-
ous, so we will focus on the other two.
(3): Let (i, j) ∈ {(0, 1), (0, 2), (1, 2)}. A ⟨σi, σj⟩-orbit consists of all the chambers
sharing the same vertex vij(C), so that by definition mij is constant on ⟨σi, σj⟩-
orbits. It is clear that C(σiσj)mij(C) = C(σiσj)rij(C)·k = C
(4): Let {i, j, k} = {0, 1, 2}. For a vertex v of colour k and i < j we define α(v) =∑
C∈DO
v∈C
(
1
mij(C)
)
.
Counting the number of chambers with a certain vertex and using the definition of
mij , we get that
182 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
v2
v1
v0,L v0,R
C1 C2
C3 C4
C5 C6 C7 C8 C9 C10
m01 m12
C1 5 6
C2 5 6
C3 5 3
C4 5 3
C5 5 3
C6 5 3
C7 5 3
C8 5 3
C9 5 3
C10 5 3
Figure 13: On the left, the double chamber patch of the lopsp-operation gyro is shown and
on the right the corresponding Delaney-Dress symbol.
α(v) =

2 if v is an inner vertex
1 if v is an outer vertex different from vi for i = 0, 1, 2
1/2 if v = v1
1/3 if v = v0
1/6 if v = v2
.
Let n be the number of vertices (and equivalently edges) in the outer face. As every
vertex of O has exactly one colour we get that:
C (DO,m01,m02,m12) =
∑
C∈DO
(
1
m01(C)
+
1
m12(C)
− 1
m02(C)
)
=
∑
C∈DO
(
1
m01(C)
+
1
m12(C)
+
1
m02(C)
)
−
∑
C∈DO
(1)
=
∑
v∈VO
α(v)− (|FO| − 1)
= (|VO| − n) · 2 + (n− 3) · 1 +
1
2
+
1
3
+
1
6
− (|FO| − 1)
= 2|VO| − |FO| − n− 1
By counting the number of directed edges associated with edges in the triangulated
disk O in two ways, we get that 2|EO| = 3(|FO| − 1) + n or equivalently |FO| =
2|EO| − 2|FO|+ 3− n. We also know that O is plane, so |VO| − |EO|+ |FO| = 2.
It follows that:
C (DO,m01,m02,m12) = 2|VO| − 2|EO|+ 2|FO| − 3 + n− n− 1 = 0
G. Brinkmann et al.: On local operations that preserve symmetries . . . 183
We will now prove the corresponding result for lopsp-operations. Let O be a lopsp-
operation and let DO be the set of chambers of O. We define the action of Σ on DO by
letting Cσi = C ′ if C and C ′ share their i-edge. For lopsp-operations there is no outer
face, so rij(C) is always
deg(vij)
2 . We define m01,m02,m12 : DO → N exactly as before:
mij(C) =

rij(C) · 2 if vij(C) = v1
rij(C) · 3 if vij(C) = v0
rij(C) · 6 if vij(C) = v2
rij(C) if vij(C) /∈ {v0, v1, v2}
Again m02(C) = 2 for all C ∈ DO. We define D(O) = (DO,m01,m02,m12) and
in Theorem 5.3 we prove that it is a Delaney-Dress symbol. The operation gyro and its
corresponding Delaney-Dress symbol are shown as an example in Figure 13. Once again,
the tiling described by the Delaney-Dress symbol is the result of applying the operation to
the hexagonal tiling of the plane. In Section 4 we named this tiling the associated tiling TO
of O. There are 2-, 3-, and 6-fold rotations at the copies of v1, v0, and v2 respectively. In
lopsp-operations there is no chamber C such that Cσi = C so there are no pure reflections
encoded in the Delaney-Dress symbol. This is the symmetry required in the geometric
definition of lopsp-operations.
Theorem 5.3. If O is a lopsp-operation, then D(O) = (DO,m01,m02,m12) is the Delaney-
Dress symbol of a tiling of the Euclidean plane.
Proof. We prove the properties in Theorem 5.1. Again, the first two are obvious.
(3): As in the proof of Theorem 5.2.
(4): Let {i, j, k} = {0, 1, 2}. For a vertex v ∈ VO of colour k and i < j we again define
α(v) =
∑
C∈DO
v∈C
(
1
mij(C)
)
.
Counting the number of chambers with a given vertex v and using the definition of
mij , we get that
α(v) =

2 if v /∈ {v0, v1, v2}
1 if v = v1
2/3 if v = v0
1/3 if v = v2
.
We can now compute C (DO,m01,m02,m12):
184 Ars Math. Contemp. 24 (2024) #P2.01 / 155–186
C (DO,m01,m02,m12) =
∑
C∈DO
(
1
m01(C)
+
1
m12(C)
− 1
m02
)
=
∑
C∈DO
(
1
m01(C)
+
1
m12(C)
+
1
m02
− 1
)
=
∑
v∈VO
α(v)− |DO|
= (|VO| − 3) · 2 +
2
3
+ 1 +
1
3
− |FO|
= 2|VO| − |FO| − 4
As O is a triangulation, we get that 2|EO| = 3|FO| and as O is plane, we have
|VO| − |EO|+ |FO| = 2. It follows that:
C (DO,m01,m02,m12) = 2|VO| − 2|EO|+ 2|FO| − 4 = 0
In Lemma 2.5 we proved that for every lsp-operation there is an equivalent lopsp-
operation Olopsp. Lemma 5.4 proves formally that the Delaney-Dress symbols of the lsp-
operation and its corresponding lopsp-operation in fact encode isomorphic tilings.
Lemma 5.4. The Delaney-Dress symbols D(O) and D(Olopsp) are Delaney-Dress sym-
bols of combinatorially isomorphic tilings.
Proof. Mapping each chamber Clopsp of D(Olopsp) onto the corresponding chamber C of
D(O), we have (in the notation of [10]) a morphism between the symbols and in the nota-
tion of [6] a Delaney map f , that is: For all k ∈ {0, 1, 2}, (i, j) ∈ {(0, 1), (0, 2), (1, 2)},
and chambers C of D(Olopsp) we have f(Cσk) = (f(C))σk and mij(C) = mij(f(C)).
The existence of such a morphism guarantees (see [6, 10]) that D(O) and D(Olopsp)
code combinatorially isomorphic tilings and that the tiling coded by D(Olopsp) can be
obtained from the tiling coded by D(O) by symmetry breaking — That is: modifying the
tiling, so that the combinatorial structure is preserved, but some metric symmetries of the
tiling are destroyed.
6 Future work
In the last section of [2] many open problems are described. They are sometimes just
formulated for lsp-operations, but are often as relevant and interesting for lopsp-operations,
so we refer the reader to [2]. A very interesting question is whether ambo is ‘essentially’
the only lsp-operation that can increase the symmetry of polyhedra, i.e. plane 3-connected
maps. More specifically: Assume that for an lsp-operation O and a polyhedron M , the
polyhedron O(M) has more symmetries than M . Is M self-dual and can O be written
as the product of ambo and other lsp-operations? For lopsp-operations this is certainly
not true. For example, applying gyro to the tetrahedron gives the dodecahedron, which
has a much larger symmetry group. Classifying lopsp-operations that can introduce new
symmetries would be an interesting problem, but maybe even more difficult than solving
the problem for lsp-operations.
G. Brinkmann et al.: On local operations that preserve symmetries . . . 185
We know that there is at least one lsp-operation (dual) that does not always preserve
3-connectivity for maps, if the face-width is at most two [1], so an obvious question is
which other operations do not always preserve 3-connectivity. This was answered for lsp-
operations in [24], where the class of such operations, called edge-breaking operations, was
characterized. Recently, these results have been extended to lopsp-operations. An article
with the new results has been submitted [25].
Another problem mentioned in [2] — the generation of lsp-operations for a given in-
flation factor — has been solved [13]. Such an algorithm not only allows the generation
of lsp-operations, but also the generation of polyhedra and other maps with some specific
symmetry groups of the embedding. For generating lopsp-operations a program has been
written very recently, but it has not been published yet.
ORCID iDs
Gunnar Brinkmann https://orcid.org/0000-0003-4168-0877
Heidi Van den Camp https://orcid.org/0000-0001-7634-3681
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.02 / 187–206
https://doi.org/10.26493/1855-3974.2896.7e6
(Also available at http://amc-journal.eu)
Algebraic degrees of 2-Cayley digraphs over
abelian groups*
Yongjiang Wu, Jing Yang, Lihua Feng †
School of Mathematics and Statistics, HNP-LAMA, Central South University,
Changsha, Hunan, 410083, P.R. China
Received 3 June 2022, accepted 22 March 2023, published online 8 September 2023
Abstract
A digraph Γ is called a 2-Cayley digraph over a group G if there exists a 2-orbit
semiregular subgroup of Aut(Γ) isomorphic to G. In this paper, we completely deter-
mine the algebraic degrees of 2-Cayley digraphs over abelian groups. This generalizes the
main results of Lu and Mönius in 2023. As applications, we consider the algebraic degrees
of Cayley digraphs over finite groups admitting an abelian subgroup of index 2. Special
attention is paid to the algebraic degrees of Cayley (di)graphs over generalized dihedral
groups, generalized dicyclic groups and semi-dihedral groups.
Keywords: Algebraic degree, 2-Cayley digraph, Abelian group.
Math. Subj. Class. (2020): 05C25, 05C50
1 Introduction
A digraph Γ consists of a finite set V (Γ) of vertices and a set E(Γ) of directed edges,
where E(Γ) ⊆ V (Γ)× V (Γ). If (u, v) ∈ E(Γ) implies (v, u) ∈ E(Γ), then Γ is said to be
undirected. For a digraph Γ on n vertices, its adjacency matrix A = (auv)n×n is defined
as
auv =
{
1, if (u, v) ∈ E(Γ),
0, otherwise.
The characteristic polynomial of Γ is the characteristic polynomial of A. The eigenvalues
of A are called the eigenvalues of Γ. The collection of eigenvalues of Γ together with their
*This research was supported by NSFC (Nos. 12271527, 12071484), Hunan Provincial Natural Science Foun-
dation (2020JJ4675, 2018JJ2479). The authors would like to express their sincere thanks to the referee for the
valuable suggestions which greatly improved the presentation of the original manuscript.
†Corresponding author.
E-mail addresses: su15273815046@163.com (Yongjiang Wu), yj1147943429@163.com (Jing Yang),
fenglh@163.com (Lihua Feng)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
188 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
multiplicities is called the spectrum of Γ, denoted by Spec(Γ). Note that A is not always
symmetric, so the eigenvalues of Γ need not be real numbers.
Let G be a finite group and S ⊆ G \ {e} , where e is the identity. The Cayley digraph
Γ = Cay(G,S) of G with respect to S is defined by V (Γ) = G and E(Γ) = {(g, sg) | g ∈
G, s ∈ S}. If S = S−1, then Γ = Cay(G,S) is called a Cayley graph. For a digraph Γ,
the set of all permutations of V (Γ) that preserve the adjacency relation of Γ forms a group,
called the automorphism group of Γ, and is denoted by Aut(Γ). By a theorem of Sabidussi
[15], a digraph Γ is a Cayley digraph over G if and only if there exists a regular subgroup
of Aut(Γ) isomorphic to G. As a generalization of Sabidussi’s Theorem [1], a digraph Γ is
called a 2-Cayley digraph over G if there exists a 2-orbit semiregular subgroup of Aut(Γ)
isomorphic to G. A 2-Cayley graph is also termed as a semi-Cayley graph in [5, 6]. A
special 2-Cayley graph is called a bi-Cayley graph in [19].
For a digraph Γ, its splitting field SF(Γ) is the smallest field extension of Q which
contains all eigenvalues of the adjacency matrix of Γ. The extension degree [SF(Γ) : Q]
is called the algebraic degree of Γ, denoted by deg(Γ). A digraph Γ is called integral
if all the eigenvalues of the adjacency matrix of Γ are integers. A digraph Γ is called
algebraically integral over a number field K if all the eigenvalues of the adjacency matrix
of Γ are algebraic integers of K. There is a close connection between the splitting field and
the algebraic integrality of a digraph. For example, for any number field K, SF(Γ) ⊆ K
if and only if Γ is algebraically integral over K. Integral graphs and algebraically integral
graphs have been extensively studied in the literature [2, 3, 4, 8, 9, 11]. In recent years,
the splitting field and algebraic degree have attracted much attention. In 2020, Mönius [13]
studied the algebraic degrees of circulant graphs Cay (Zp, S) for a prime number p. In
2022, Mönius [14] generalized those results in [13] by determining the splitting fields and
the algebraic degrees of circulant graphs Cay (Zn, S) for arbitrary n. Based on Mönius’s
work, in 2022, Huang et al. [18] determined the splitting fields and algebraic degrees of
mixed Cayley graphs over abelian groups. Lu et al. [12] determined the splitting fields
of Cayley graphs over abelian groups and dihedral groups. They also gave bounds for the
algebraic degrees of Cayley graphs over dihedral groups. Also in 2022, Sripaisan et al.
[16] studied the algebraic degrees of Cayley hypergraphs. For more details, one may refer
to the comprehensive survey [10] in this subject.
In this paper, inspired by the above mentioned results, we completely determine the
splitting fields and algebraic degrees of 2-Cayley digraphs over abelian groups in Section 3,
which generalizes the main results of [12]. From computational viewpoints, we also derive
sharp upper and lower bounds for their algebraic degrees. As applications, in Section 4, we
consider the algebraic degrees of Cayley digraphs over finite groups admitting an abelian
subgroup of index 2. Furthermore, we consider the algebraic degrees of Cayley graphs
over generalized dihedral groups and generalized dicyclic groups, and get improved upper
bounds. Finally, we determine the algebraic degrees of Cayley digraphs over semi-dihedral
groups.
2 Preliminaries
Let G be a finite group. A representation of G is a homomorphism ρ : G → GL(V )
for some n-dimensional vector space over the complex field C, where GL(V ) denotes
the group of automorphisms of V . The dimension of V is called the degree of ρ . Two
representations ρ1 and ρ2 of G on V1 and V2 respectively are equivalent if there is an
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 189
isomorphism T : V1 → V2 such that Tρ1(g) = ρ2(g)T for all g ∈ G.
Let ρ : G → GL(V ) be a representation. The character χρ : G → C of ρ is defined
by setting χρ(g) = Tr(ρ(g)) for g ∈ G, where Tr(ρ(g)) is the trace of the representation
matrix of ρ(g) with respect to a specified basis of V . By the degree of χρ we mean the
degree of ρ, which is simply χρ(1). If W is a ρ(g)-invariant subspace of V for each g ∈ G,
then we call W a ρ(G)-invariant subspace of V . If the only ρ(G)-invariant subspace of V
are {0} and V , we call ρ an irreducible representation of G, and the corresponding charac-
ter χρ an irreducible character of G. We denote by IRR(G) and Irr(G) the complete set
of non-equivalent irreducible representations of G and the complete set of non-equivalent
irreducible characters of G, respectively.
For any subset X ⊆ G, we denote by δX = (δg)g∈G the characteristic vector of X over
G, where δg = 1 if g ∈ X and δg = 0 if g /∈ X . For any multi-subset X ⊆ G, we denote
by δ′X =
(
δ′g
)
g∈G the characteristic vector of X over G, where δ
′
g = k if g appears k times
in X and δ′g = 0 if g /∈ X . Throughout this paper, we use X = [x | x ∈ X] to denote
the multi-set X , and φ(n) to denote the Euler totient function of a natural number n (it is
the number of the positive integers which are smaller than n and coprime to n). Firstly, we
state an equivalent definition of 2-Cayley digraphs.
Lemma 2.1 ([1]). A digraph Γ is a 2-Cayley digraph over G if and only if there exist
subsets Tij of G, where 1 ⩽ i, j ⩽ 2, such that Γ is isomorphic to a digraph Υ with
V (Υ) = G× {1, 2}, E(Υ) =
⋃
1⩽i,j⩽2
{((g, i), (tg, j)) | g ∈ G and t ∈ Tij} .
By Lemma 2.1, a 2-Cayley digraph is characterized by a group G and four subsets
Tij of G. Thus we denote a 2-Cayley digraph with respect to four subsets Tij by Γ =
Cay (G;Tij | 1 ⩽ i, j ⩽ 2). Note that V (Γ) = G × {1, 2}, (g, i) ∼ (h, j) if and only if
hg−1 ∈ Tij , and Γ is undirected if and only if for all 1 ⩽ i, j ⩽ 2, Tij = T−1ji . Note also
that Γ is a digraph without loops if and only if Tii ⊆ G\{e}, for all 1 ⩽ i ⩽ 2.
Let ωn = exp
(
2πi
n
)
be the primitive n-th root of unity. We consider an abelian group
G of order n. It is well known that
G ∼= Zn1 ⊕ · · · ⊕ Znr ,
where n =
∏r
i=1 ni, and ni is a prime power for 1 ≤ i ≤ r. Without loss of generality, we
assume that G = Zn1 ⊕ · · · ⊕ Znr and 0 = (0, . . . , 0) ∈ G is the identity of G.
Lemma 2.2 ([17]). Let G = Zn1⊕· · ·⊕Znr be an abelian group of order n. Then Irr(G) =
{χl | l ∈ G}, where χl(g) =
∏r
i=1 ω
ligi
ni for all l = (l1, . . . , lr) , g = (g1, . . . , gr) ∈ G,
and ωni = exp(
2πi
ni
).
For simplicity, for any (multi-)subset S of G, we denote
χl(S) =
∑
s∈S
χl(s).
Arezoomand [1] obtained the following result.
Lemma 2.3 ([1]). Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an
abelian group G = Zn1 ⊕ · · · ⊕ Znr of order n. Then Γ has eigenvalues
χl(T11) + χl(T22)±
√
(χl(T11)− χl(T22))2 + 4χl(T21)χl(T12)
2
, l ∈ G.
190 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
Let K be a field. In what follows, we will refer to the subgroup
K×2 =
{
x2 : x ∈ K
}
⊂ K×,
where K× = K \ {0}. More precisely, we shall encounter quite often the quotient
K×/K×2. The image of x ∈K×in K×/K×2 will be denoted by [x]K .
Lemma 2.4 ([7, Corollary 1.23]). Suppose K is a field containing a primitive 2-th root of
unity, and let F = K
[√
a1, . . . ,
√
ak
]
, where ai ∈ K. Then Gal(F/K) is isomorphic to
the subgroup of K×/K×2 generated by [a1]K , . . . , [ak]K .
3 2-Cayley digraphs over abelian groups
In this section, we always assume that G = Zn1 ⊕· · ·⊕Znr is an abelian group of order n.
Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over G. For any two subsets X
and Y of G, we define the multi-set X + Y = [x+ y | x ∈ X, y ∈ Y ]. For any (multi-)set
X and k ∈ N, k ∗X denotes the multi-set in which each element of X appears k times. For
example, if X = [1, 1, 2, 2, 2, 3, 4] and k = 2, then k ∗X = 4 ∗ {1} ∪ 6 ∗ {2} ∪ 2 ∗ {3, 4},
with duplicate elements allowed in the union.
For two multi-sets U = k1 ∗ {g1} ∪ k2 ∗ {g2} ∪ . . . ∪ ks ∗ {gs} and V = q1 ∗ {g1} ∪
q2 ∗ {g2} ∪ . . . ∪ qt ∗ {gt} ∪ ks+1 ∗ {gs+1} ∪ . . . ∪ ks+m ∗ {gs+m}, where ki ≥ qi, s > t
and gi, 1 ≤ i ≤ s + m are pairwise distinct, we define U \ V = (k1 − q1) ∗ {g1} ∪
(k2 − q2) ∗ {g2} ∪ . . . ∪ (kt − qt) ∗ {gt} ∪ kt+1 ∗ {gt+1} ∪ . . . ∪ ks ∗ {gs}, V \ U =
ks+1 ∗ {gs+1} ∪ . . . ∪ ks+m ∗ {gs+m}.
Using the symbols in Lemma 2.3, we let
I1 = [t | t ∈ T11 or t ∈ T22] ,
I2 = [t | t ∈ (T11 + T11) or t ∈ (T22 + T22) or t ∈ 4 ∗ (T12 + T21)],
I3 = [t | t ∈ 2 ∗ (T11 + T22)],
where I1, I2 and I3 are multi-sets. For example, for the group G = Z4, if T11 = {1, 2},
T12 = {1}, T21 = {2} and T22 = {3}, then I1 = {1, 2, 3}, I2 = 6 ∗ {3} ∪ 2 ∗ {2} ∪ {0}
and I3 = 2 ∗ {0, 1}.
By Lemma 2.3, we have the following result.
Lemma 3.1. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. Then Γ has eigenvalues
χl(I1)±
√
χl(I2 \ I3)− χl(I3 \ I2)
2
, l ∈ G,
where I1, I2, I3 are described as above.
Proof. Firstly, we have
χl(T11) + χl(T22) = χl(I1).
In addition,
(χl(T11)− χl(T22))2 + 4χl(T21)χl(T12)
=χl(T11 + T11) + χl(T22 + T22) + χl (4 ∗ (T12 + T21))− χl (2 ∗ (T11 + T22))
=χl(I2)− χl(I3)
=χl(I2 \ I3)− χl(I3 \ I2),
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 191
so the result follows from Lemma 2.3.
Using the symbols in Lemma 3.1, for l ∈ G, let
βl = χl(I1) and γl = χl(I2 \ I3)− χl(I3 \ I2). (3.1)
As βl, γl ∈ Q (ωn) , where n = |G|, without loss of generality, we assume that K is a field
such that Q ⊆ K ⊆ Q (ωn). Therefore, Gal (Q (ωn) /K)) ≤ Gal (Q (ωn) /Q) ∼= Z∗n =
{k ∈ Zn | gcd(k, n) = 1}. Let
η : Gal (Q (ωn) /Q) → Z∗n
be the isomorphism such that σ (ωn) = ω
η(σ)
n , where σ ∈ Gal (Q (ωn) /Q). Let
H = η (Gal (Q (ωn) /K)) .
Then H is a subgroup of Z∗n. We consider the action of Z∗n on G = Zn1 ⊕ · · · ⊕ Znr by
setting kg = k (g1, . . . , gr) = (kg1, . . . , kgr) for any k ∈ Z∗n and g ∈ G. Then
σ
(
ωlini
)
= σ
(
ωnli/nin
)
= ωη(σ)·nli/nin = ω
η(σ)li
ni ,
where li ∈ Zni(1 ≤ i ≤ r). Note that for any σ ∈ Gal (Q (ωn) /Q), we have
σ (βl) = σ (χl(I1)) = σ
(∑
t∈I1
r∏
i=1
ωlitini
)
=
∑
t∈I1
r∏
i=1
σ
(
ωlitini
)
=
∑
t∈I1
r∏
i=1
ωη(σ)litini = χl(η(σ)I1),
where η(σ)I1 = {(η(σ)t1, . . . , η(σ)tr) | (t1, . . . , tr) ∈ I1}. Similarly, we have
σ (γl) = σ (χl(I2 \ I3)− χl(I3 \ I2))
= χl (η(σ)(I2 \ I3))− χl (η(σ)(I3 \ I2)) ,
where I2 \ I3 and I3 \ I2 are multi-sets as stated in Lemma 3.1.
We first prove the following results.
Proposition 3.2. For the symbols in (3.1), we have βl ∈ K for all l ∈ G if and only if
hI1 = I1 for all h ∈ H , where H = η (Gal (Q (ωn) /K)).
Proof. Assume that hI1 = I1 for all h ∈ H . Then for any σ ∈ Gal (Q (ωn) /K), we have
η(σ) ∈ H . Thus for any l ∈ G, we have
σ (βl) = χl(η(σ)I1) = χl(I1) = βl.
It follows that βl ∈ K for all l ∈ G.
Conversely, assume that βl ∈ K for all l ∈ G. For any h ∈ H , there exists some
σ ∈ Gal (Q (ωn) /K) such that η(σ) = h. Then
χl(hI1) = χl(η(σ)I1) = σ (βl) = βl = χl(I1).
192 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
Let M = (χl (g))l,g∈G. We get
Mδ′hI1 = Mδ
′
I1 .
Note that M is invertible by the orthogonal relations of irreducible characters of G. So we
have
δ′hI1 = δ
′
I1 .
This implies hI1 = I1. Since h is arbitrary, the result follows.
Proposition 3.3. For the symbols in (3.1), we have γl ∈ K for all l ∈ G if and only if
h(I2 \ I3) = I2 \ I3, h(I3 \ I2) = I3 \ I2 for all h ∈ H , where H = η (Gal (Q (ωn) /K)).
Proof. Assume that h(I2 \ I3) = I2 \ I3, h(I3 \ I2) = I3 \ I2 for all h ∈ H . Then for any
σ ∈ Gal (Q (ωn) /K), we have η(σ) ∈ H . Thus, for any l ∈ G, we have
σ (γl) = χl (η(σ)(I2 \ I3))− χl (η(σ)(I3 \ I2)) = χl(I2 \ I3)− χl(I3 \ I2) = γl.
It follows that γl ∈ K for all l ∈ G.
Conversely, assume that γl ∈ K for all l ∈ G. For any h ∈ H , there exists some
σ ∈ Gal (Q (ωn) /K) such that η(σ) = h. Then
χl (η(σ)(I2 \ I3))− χl (η(σ)(I3 \ I2)) = σ (γl) = γl = χl(I2 \ I3)− χl(I3 \ I2).
This means that
χl (h(I2 \ I3))− χl (h(I3 \ I2)) = χl(I2 \ I3)− χl(I3 \ I2).
Let M = (χl (g))l,g∈G. We get
Mδ′h(I2\I3) −Mδ
′
h(I3\I2) = Mδ
′
I2\I3 −Mδ
′
I3\I2 .
Note that M is invertible and I2 \I3 is disjoint with I3 \I2. So we have h(I2 \I3) = I2 \I3
and h(I3 \ I2) = I3 \ I2. As h is arbitrary, the result follows.
Note that β0, γ0 ∈ Z, so we let
L = K = Q (βl, γl | l ∈ G \ {0}) (3.2)
and
H ′ = {h ∈ Z∗n | hI1 = I1, h(I2 \ I3) = I2 \ I3, h(I3 \ I2) = I3 \ I2} . (3.3)
Then we have the following result.
Proposition 3.4. Using the symbols in (3.2) and (3.3), we have H ′ = η (Gal (Q (ωn) /L)).
Proof. By Propositions 3.2 and 3.3, it is clear that
η (Gal (Q (ωn) /L)) ⊆ H ′.
Now we prove
H ′ ⊆ η (Gal (Q (ωn) /L)) .
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 193
For each h′ ∈ H ′, let σ = η−1(h′). It follows that h′ = η(σ). For any l ∈ G, we have
σ (βl) = χl(η(σ)I1) = χl(h
′I1) = χl(I1) = βl.
Similarly, for any l ∈ G,
σ (γl) = χl (η(σ)(I2 \ I3))− χl (η(σ)(I3 \ I2)) = χl(I2 \ I3)− χl(I3 \ I2) = γl.
Hence
σ ∈ Gal (Q (ωn) /L) and h′ = η(σ) ∈ η (Gal (Q (ωn) /L)) .
Thus the result follows.
Since H ′ is a subgroup of Z∗n, by Proposition 3.4, we have
L = Q (ωn)η
−1(H′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′)}. (3.4)
Considering H ′ acting on G, assume that H ′g(1), H ′g(2), . . . ,H ′g(k) are all distinct orbits
of H ′ on G, where g(i) ∈ G. Let
C =
{
g(i) | G ∩H ′g(i) ̸= ∅
}
. (3.5)
Let M be the subgroup of L×/L×2 generated by all [γl]L for l ∈ C. Explicitly,
M = ⟨[γl]L | l ∈ C⟩ . (3.6)
Now we are ready to prove our main result.
Theorem 3.5. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. Then the splitting field of Γ is L
(√
γl | l ∈ C
)
and the algebraic
degree of Γ satisfies
deg(Γ) =
φ(n)|M |
|H ′|
,
where γl, H ′, L, C,M are given in (3.1) and (3.3) – (3.6), respectively.
Proof. If a, b are in the same orbit H ′g(i), then there exists h ∈ H ′ such that b = ha. It
follows that
γb = χb(I2 \ I3)− χb(I3 \ I2) = χha(I2 \ I3)− χha(I3 \ I2)
= χa (h(I2 \ I3))− χa (h(I3 \ I2)) = χa(I2 \ I3)− χa(I3 \ I2)
= γa.
Therefore, there are at most |C| different elements in {γl | l ∈ G}.
Set F = L
(√
γl | l ∈ C
)
. Note that F = Q
(
βl +
√
γl, βl −
√
γl | l ∈ G
)
. So the first
assertion follows. By Lemma 2.4,
deg(Γ) = [F : Q] = [F : L][L : Q] =
[Q (ωn) : Q] [F : L]
[Q (ωn) : L]
=
φ(n)|M |
|H ′|
.
This completes the proof.
194 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
It is not easy to calculate |M |, but apparently 1 ≤ |M | ≤ 2|C|, thus we have
Corollary 3.6. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. Then the algebraic degree of Γ satisfies
φ(n)
|H ′|
≤ deg(Γ) ≤ φ(n)2
|C|
|H ′|
,
where H ′, C are given in (3.3) and (3.5), respectively.
Remark 3.7. Theorem 3.5 and Corollary 3.6 still hold for a 2-Cayley graph. Indeed, we
just need to restrict Tij = T−1ji for all 1 ⩽ i, j ⩽ 2, and modify the associated multi-sets
I1, I2 and I3.
The next two examples tell us that both the lower and upper bound in Corollary 3.6 are
sharp.
Example 3.8. Let Γ = Cay (Z3, Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley graph over G = Z3. Let
T11 = T22 = {1, 2} and T12 = {1} and T21 = {2}. Then I1 = 2∗{1, 2}, I2 \I3 = 4∗{0}
and I3 \ I2 = ∅. It follows that H ′ = {1, 2} = Z∗3, L = Q and γl = 4 for all l ∈ Z3. Thus
|M | = 1 and deg(Γ) = φ(3)|H′| = 1. In fact, Spec(Γ) = 2 ∗ {−2, 0} ∪ {1, 3}.
Example 3.9. Let Γ = Cay (Z4, Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over G = Z4.
Let T11 = {1, 2} and T12 = {1}. Let T21 = {2} and T22 = {3}. Then I1 = {1, 2, 3},
I2 \ I3 = 6 ∗ {3} ∪ 2 ∗ {2} and I3 \ I2 = 2 ∗ {1} ∪ {0}. It follows that H ′ = {1}
and C = Z4. By Corollary 3.6, deg(Γ) ≤ 25 = 32. In fact, L = Q (i) and F =
L
(√
5,
√
−8i− 3,
√
−3,
√
8i− 3
)
. Obviously, deg(Γ) = 32.
Observe that L = Q if and only if |H ′| = φ(n), as an application of Theorem 3.5, the
next corollary provides a class of integral 2-Cayley digraphs over abelian groups.
Corollary 3.10. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. If H ′ = Z∗n and γl is a square of an integer for each l ∈ C, where
γl, H
′, C are given in (3.1), (3.3) and (3.5), respectively, then Γ is integral.
Sometimes, we need not to compute |M | in Theorem 3.5.
Corollary 3.11. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. If T11 = T22 and T12 = T−112 = T21, then the splitting field of Γ satis-
fies
SF(Γ) = Q (ωn)η
−1(H′′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′′)},
the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)
|H ′′|
,
where H ′′ = {h ∈ Z∗n | hT11 = T11, hT12 = T12}.
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 195
Proof. Since I1 = [t | t ∈ 2 ∗ T11], I2 \ I3 =
[
t | t ∈ 4 ∗ (T12 + T−112 )
]
and I3 \ I2 = ∅,
we have γl = 4χl(T12 + T−112 ) = 4|χl(T12)|2 = 4χl(T12)2. Note that T12 = T
−1
12 . So
χl(T12) is a real number. It follows that
√
γl = 2χl(T12) or
√
γl = −2χl(T12).
The rest of the proof is similar to that of Theorem 3.5.
Corollary 3.12. Let Γ = Cay (G,Tij | 1 ⩽ i, j ⩽ 2) be a 2-Cayley digraph over an abelian
group G of order n. If T11 = T22, T12 = T−112 = T21, and hT11 = T11, hT12 = T12 for all
h ∈ Z∗n, then Γ is integral.
4 Some applications
4.1 Cayley digraphs over groups admitting an abelian subgroup of index 2
A Cayley digraph over a finite group G with a subgroup of index 2 is a 2-Cayley digraph,
as the following result shows.
Lemma 4.1 ([1]). Let Γ = Cay(G,S) be a Cayley (di)graph. Suppose that there exists
a subgroup N of G with index 2. If {x1, x2} is a left transversal to N in G, then Γ ∼=
Cay (N,Sij | 1 ⩽ i, j ⩽ 2), where Sij = {a ∈ N | x−1j axi ∈ S} = N ∩ xjSx
−1
i .
Let A be a finite abelian group of order n ≥ 3. Let f ∈ Aut(A) be of order 2. Let
y ∈ A be such that f(y) = y. Let G be a non-abelian finite group admitting an abelian
subgroup A of index 2. Then G admits a presentation
G =
〈
A, x | x2 = y, xax−1 = f(a), a ∈ A
〉
.
Observe that G = A ∪ xA and B =
{
f(a)a−1 | a ∈ A
}
is a subgroup of A. In particular,
if f(a) = a−1 for a ∈ A, then B = A2 and y2 = e, where e is the identity of A. If y = e,
then G is the generalized dihedral group Dih(A), with the presentation
Dih(A) =
〈
A, x | x2 = e, xax−1 = a−1, a ∈ A
〉
.
If y ̸= e (and so n = |A| is even), then G is the generalized dicyclic group Dic(A, y), with
the presentation
Dic(A, y) =
〈
A, x | x2 = y, xax−1 = a−1, a ∈ A
〉
.
As the group operation here is multiplication, we assume that A = ⟨a1⟩n1⊗· · ·⊗⟨ar⟩nr
and Irr(A) = {χl | (al11 , . . . , alrr ) ∈ A}, where l = (l1, . . . , lr). In this subsection, we
always assume that G is a group admitting an abelian subgroup A of order n and of index 2.
As an application of Theorem 3.5, we consider the algebraic degree of the Cayley digraph
Γ = Cay(G,S). Note that A ∼= A′ = Zn1 ⊕ · · · ⊕ Znr . It is worth pointing out that the
group operation here should correspond to the addition in Section 3.
By Lemmas 2.3 and 4.1, we get the following result.
Lemma 4.2. Let Γ = Cay(G,S) be a Cayley digraph and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr be
an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. Then Γ
has eigenvalues
χl(T11) + χl(T22)±
√
(χl(T11)− χl(T22))2 + 4χl(T21)χl(T12)
2
, l = (l1, . . . , lr) ∈ A′,
196 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
where n =
∏r
i=1 ni, Tij =
{
t = (t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xjSx−1i
}
and A′ =
Zn1 ⊕ · · · ⊕ Znr .
Using the symbols in Lemma 4.2, in a similar way as in Section 3, we define
I1 = [t | t ∈ T11 or t ∈ T22] ,
I2 = [t | t ∈ (T11 + T11) or t ∈ (T22 + T22) or t ∈ 4 ∗ (T12 + T21)],
I3 = [t | t ∈ 2 ∗ (T11 + T22)].
Let
βl = χl(I1) and γl = χl(I2 \ I3)− χl(I3 \ I2). (4.1)
Let η : Gal (Q (ωn) /Q) → Z∗n be the isomorphism such that σ (ωn) = ω
η(σ)
n , where
σ ∈ Gal (Q (ωn) /Q). Let
L = Q (βl, γl | l ∈ A′ \ {0}) (4.2)
and
H ′ = {h ∈ Z∗n | hI1 = I1, h(I2 \ I3) = I2 \ I3, h(I3 \ I2) = I3 \ I2} . (4.3)
Since Γ ∼= Cay (A′, Tij | 1 ⩽ i, j ⩽ 2), by Proposition 3.4, we have the following
result.
Proposition 4.3. Using the symbols in (4.2) and (4.3), we have H ′ = η (Gal (Q (ωn) /L)).
Now we consider H ′ acting on A′ = Zn1⊕· · ·⊕Znr . Assume that H ′a(1), H ′a(2), . . . ,
H ′a(k) are all distinct orbits of H ′ on A′, where a(i) ∈ A′. Let
C =
{
a(i) | A′ ∩H ′a(i) ̸= ∅
}
(4.4)
and
M = ⟨[γl]L | l ∈ C⟩ . (4.5)
Since Γ ∼= Cay (A′, Tij | 1 ⩽ i, j ⩽ 2), using the conclusions in Section 3, we imme-
diately get the following results.
Theorem 4.4. Let Γ = Cay(G,S) be a Cayley digraph, and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr
be an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. Then
the splitting field of Γ is L
(√
γl | l ∈ C
)
and the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)|M |
|H ′|
,
where L = Q (ωn)η
−1(H′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′)} and
γl, H
′, C,M are given in (4.1) and (4.3) – (4.5), respectively.
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 197
Corollary 4.5. Let Γ = Cay(G,S) be a Cayley digraph, and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr
be an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. Then
the algebraic degree of Γ satisfies
φ(n)
|H ′|
≤ deg(Γ) ≤ φ(n)2
|C|
|H ′|
,
where H ′, C are given in (4.3) and (4.4), respectively.
Corollary 4.6. Let Γ = Cay(G,S) be a Cayley digraph, and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr
be an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. If
H ′ = Z∗n and γl is a square of an integer for each l ∈ C, where γl, H ′, C are given in
(4.1), (4.3) and (4.4), respectively, then Γ is integral.
Corollary 4.7. Let Γ = Cay(G,S) be a Cayley digraph, and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr
be an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. Let
Tij =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xjSx−1i
}
. If T11 = T22 and T12 = T−112 = T21,
then the splitting field of Γ satisfies
SF(Γ) = Q (ωn)η
−1(H′′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′′)},
the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)
|H ′′|
,
where H ′′ = {h ∈ Z∗n | hT11 = T11, hT12 = T12}.
Corollary 4.8. Let Γ = Cay(G,S) be a Cayley digraph, and A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr
be an abelian subgroup of G of order n and of index 2 with left transversal {x1, x2}. Let
Tij =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xjSx−1i
}
. If T11 = T22, T12 = T−112 = T21, and
hT11 = T11, hT12 = T12 for all h ∈ Z∗n, then Γ is integral.
4.2 Cayley graphs over generalized dihedral groups
In the following two subsections, we consider Cayley graphs but not digraphs. The gener-
alized dihedral group Dih(A) is given by the following presentation
Dih(A) =
〈
A, x | x2 = e, xax−1 = a−1, a ∈ A
〉
.
Let Γ = Cay(Dih(A), S) be a Cayley digraph. Using the symbols in Subsection 4.1,
note that A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr , and |A| = n, so |Dih(A)| = 2n. Without loss of
generality, let x1 = e and x2 = x. Then
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ S
}
,
T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
,
T21 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ Sx
}
,
T22 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xSx
}
.
(4.6)
For the algebraic degree of the digraph Γ, we just need to replace Tij given in Subsec-
tion 4.1 with Tij given in (4.6), so we omit the details here.
198 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
We are now interested in the algebraic degree of the undirected Cayley graph Γ =
Cay(Dih(A), S). Using the symbols in (4.6), as S = S−1, we have T−122 = T22 = T11 =
T−111 and T
−1
12 = T21. Let t = (t1, . . . , tr). In a similar way as in Subsection 4.1, we define
I1 = [t | t ∈ T11 or t ∈ T22] ,
I2 = [t | t ∈ (T11 + T11) or t ∈ (T22 + T22) or t ∈ 4 ∗ (T12 + T21)],
I3 = [t | t ∈ 2 ∗ (T11 + T22)].
It follows that I1 = [t | t ∈ 2 ∗ T11], I2 \ I3 =
[
t | t ∈ 4 ∗ (T12 + T−112 )
]
and I3 \ I2 = ∅.
In fact, by Lemma 4.2, the eigenvalues of the Cayley graph Γ = Cay(Dih(A), S) are
χl(T11)± |χl(T12)| , l ∈ A′,
where A′ = Zn1 ⊕ · · · ⊕ Znr . Note that |χl(T12)| =
√
χl(T12)χl(T
−1
12 ) =√
χl(T12 + T
−1
12 ), the multi-sets I1 and I2 \I3 can be reduced to I ′1 = T11 and (I2 \I3)′ =[
t | t ∈ T12 + T−112
]
.
Let
βl = χl(I
′
1) and γl = χl((I2 \ I3)′). (4.7)
Let η : Gal (Q (ωn) /Q) → Z∗n be the isomorphism such that σ (ωn) = ω
η(σ)
n , where
σ ∈ Gal (Q (ωn) /Q). Let
L = Q (βl, γl | l ∈ A′ \ {0}) (4.8)
and
H ′ = {h ∈ Z∗n | hI ′1 = I ′1, h(I2 \ I3)′ = (I2 \ I3)′} . (4.9)
Note that I1 = 2 ∗ I ′1 and I2 \ I3 = 4 ∗ (I2 \ I3)′. So we have the following result by
Proposition 4.3.
Proposition 4.9. Using the symbols in (4.8) and (4.9), we have H ′ = η (Gal (Q (ωn) /L)).
Similarly, we consider H ′ acting on A′. Assume that H ′a(1), H ′a(2), . . . H ′a(k) are
all distinct orbits of H ′ on A′, where a(i) ∈ A′. Let B = {x ∈ A′ | 2x = 0} and
A′ = B ∪ E ∪ E−1, where B,E,E−1 are disjoint. Let
C ′ =
{
a(i) | (B ∪ E) ∩H ′a(i) ̸= ∅
}
(4.10)
and
M = ⟨[γl]L | l ∈ C ′⟩ . (4.11)
Then the following results hold.
Theorem 4.10. Let Γ = Cay(Dih(A), S) be a Cayley graph over the generalized dihe-
dral group Dih(A) of order 2n. Then the splitting field of Γ is L
(√
γl | l ∈ C ′
)
and the
algebraic degree of Γ satisfies
deg(Γ) =
φ(n)|M |
|H ′|
,
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 199
where L = Q (ωn)η
−1(H′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′)} and
γl, H
′, C ′,M are given in (4.7) and (4.9) – (4.11), respectively.
Proof. Since ((I2 \ I3)′)−1 = (I2 \ I3)′, it follows that
γl = χl((I2 \ I3)′) = χl(((I2 \ I3)′)−1) = χ−l((I2 \ I3)′) = γ−l.
Then the result follows from Theorem 4.4.
Corollary 4.11. Let Γ = Cay(Dih(A), S) be a Cayley graph over the generalized dihedral
group Dih(A) of order 2n. Then the algebraic degree of Γ satisfies
φ(n)
|H ′|
≤ deg(Γ) ≤ φ(n)2
|C′|
|H ′|
,
where H ′, C ′ are given in (4.9) and (4.10), respectively.
Corollary 4.12. Let Γ = Cay(Dih(A), S) be a Cayley graph over the generalized dihedral
group Dih(A) of order 2n. If H ′ = Z∗n and γl is a square of an integer for each l ∈ C ′,
where γl, H ′, C ′ are given in (4.7), (4.9) and (4.10), respectively, then Γ is integral.
Corollary 4.13. Let Γ = Cay(Dih(A), S) be a Cayley graph over the generalized dihedral
group Dih(A) of order 2n. Let
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ S
}
and T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
.
If T12 = T−112 , then the splitting field of Γ satisfies
SF(Γ) = Q (ωn)η
−1(H′′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′′)},
the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)
|H ′′|
,
where H ′′ = {h ∈ Z∗n | hT11 = T11, hT12 = T12}.
Corollary 4.14. Let Γ = Cay(Dih(A), S) be a Cayley graph over the generalized dihedral
group Dih(A) of order 2n. Let
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ S
}
and T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
.
If T12 = T−112 and for all h ∈ Z∗n, hT11 = T11 and hT12 = T12, then Γ is integral.
In particular, let Dih(A) = D2n = ⟨a, b | an = b2 = e, bab = a−1⟩ be the dihedral
group of order 2n. Then A′ = Zn, I ′1 = T11 = {t | at ∈ S}, T12 = {t | bat ∈ S} and
(I2 \ I3)′ =
[
t | t ∈ T12 + T−112
]
. Note that
βl = χl(I
′
1) and γl = χl((I2 \ I3)′), (4.12)
where χl(t) = ωltn and 0 ≤ l ≤ n− 1. Furthermore,
L = Q (βl, γl | 1 ≤ l ≤ n− 1) .
We first try to simplify the expression of L.
200 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
Lemma 4.15. Let K be a field such that Q ⊆ K ⊆ Q (ωn). If β1, γ1 ∈ K, then βl, γl ∈ K
for 1 ≤ l ≤ n− 1.
Proof. For 1 ≤ l ≤ n − 1, let σl : Q (ωn) → Q (ωn) be defined by σl (ωn) = ωln.
It is clear that σl is a homomorphism and βl = σl (β1) , γl = σl (γ1). Thus, for any
σ ∈ Gal (Q (ωn) /K), we have
σ (βl) = σ (σl (β1)) = σ
σl
∑
t∈I′1
ωtn
 = ∑
t∈I′1
ωη(σ)tln = σl (σ (β1)) = σl (β1) = βl.
Similarly, σ (γl) = γl. Therefore, βl, γl ∈ K.
By Lemma 4.15, we have
L = Q (β1, γ1) .
Let
H ′ = {h ∈ Z∗n | hI ′1 = I ′1, h(I2 \ I3)′ = (I2 \ I3)′} . (4.13)
Then H ′ = η (Gal (Q (ωn) /L)). Note that
C ′ =
{
a(i) | {0, 1, . . . , ⌊n/2⌋} ∩H ′a(i) ̸= ∅
}
(4.14)
and
M = ⟨[γl]L | l ∈ C ′⟩ , (4.15)
where H ′a(1), H ′a(2), . . . H ′a(k) are all distinct orbits of H ′ on Zn. Consequently, we
have the following corollaries.
Corollary 4.16. Let Γ = Cay(D2n, S) be a Cayley graph over the dihedral group D2n.
Then the splitting field of Γ is L
(√
γl | l ∈ C ′
)
and the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)|M |
|H ′|
,
where L = Q (ωn)η
−1(H′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′)} and
γl, H
′, C ′,M are given in (4.12) – (4.15), respectively.
Corollary 4.17. Let Γ = Cay(D2n, S) be a Cayley graph over the dihedral group D2n.
Then the algebraic degree of Γ satisfies
φ(n)
|H ′|
≤ deg(Γ) ≤ φ(n)2
|C′|
|H ′|
,
where H ′, C ′ are given in (4.13) and (4.14), respectively.
Corollary 4.18. Let Γ = Cay(D2n, S) be a Cayley graph over the dihedral group D2n.
Let T11 = {t | at ∈ S} and T12 = {t | bat ∈ S}. If T12 = T−112 , then the splitting field of
Γ satisfies
SF(Γ) = Q (ωn)η
−1(H′′)
= {x ∈ Q (ωn) | σ(x) = x for all σ ∈ η−1(H ′′)},
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 201
the algebraic degree of Γ satisfies
deg(Γ) =
φ(n)
|H ′′|
,
where H ′′ = {h ∈ Z∗n | hT11 = T11, hT12 = T12}.
There are results similar to Corollaries 4.12 and 4.14 as well, we omit them here. We
end this subsection with the following example.
Example 4.19. Let D16 = ⟨a, b | a8 = b2 = e, bab = a−1⟩ be the dihedral group of
order 16 and S =
{
a, a7, b
}
. We consider the algebraic degree of Γ = Cay(D16, S).
Then T11 = {1,−1} and T12 = {0} = T−112 . It follows that H ′′ = {1,−1} ≤ Z∗8. By
Corollary 4.18, SF(Γ) = Q (ω8)η
−1(H′′)
= Q(
√
2) and deg(Γ) = φ(8)|H′′|=2. In fact,
Spec(Γ) = 2 ∗ {
√
2 + 1,
√
2− 1,−
√
2 + 1,−
√
2− 1} ∪ 3 ∗ {1,−1} ∪ {−3, 3}.
4.3 Cayley graphs over generalized dicyclic groups
For the generalized dicyclic group Dic(A, y), it has the following presentation
Dic(A, y) =
〈
A, x | x2 = y, xax−1 = a−1, a ∈ A
〉
.
We put our focus on the algebraic degree of the Cayley graph Γ = Cay(Dic(A, y), S).
Using the symbols in Subsection 4.1, since A = ⟨a1⟩n1 ⊗ · · · ⊗ ⟨ar⟩nr , and |A| = n is
even, say n = 2m, then |Dic(A, y)| = 4m. Let x1 = e and x2 = x. Then
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ S
}
,
T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
,
T21 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ Sx−1
}
,
T22 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xSx−1
}
.
Since S = S−1, we have T−122 = T22 = T11 = T
−1
11 and T
−1
12 = T21. Let t =
(t1, . . . , tr). By similar arguments as those in Subsection 4.2, we just need to consider
I ′1 = T11 and (I2 \ I3)′ =
[
t | t ∈ T12 + T−112
]
. Let
βl = χl(I
′
1) and γl = χl((I2 \ I3)′). (4.16)
Let η : Gal (Q (ω2m) /Q) → Z∗2m be the isomorphism such that σ (ω2m) = ω
η(σ)
2m , where
σ ∈ Gal (Q (ω2m) /Q).
Let L = Q (βl, γl | l ∈ A′ \ {0}) and
H ′ = {h ∈ Z∗2m | hI ′1 = I ′1, h(I2 \ I3)′ = (I2 \ I3)′} . (4.17)
By Proposition 4.3, we have H ′ = η (Gal (Q (ω2m) /L)).
Also, we consider H ′ acting on A′ = Zn1 ⊕· · ·⊕Znr . Assume that H ′a(1), H ′a(2), . . .
H ′a(k) are all distinct orbits of H ′ on A′, where a(i) ∈ A′. Let B = {x ∈ A′ | 2x = 0}
and A′ = B ∪ E ∪ E−1, where B,E,E−1 are disjoint. Let
C ′ =
{
a(i) | (B ∪ E) ∩H ′a(i) ̸= ∅
}
(4.18)
202 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
and
M = ⟨[γl]L | l ∈ C ′⟩ . (4.19)
In a similar way as in Theorem 4.10, we get the following result.
Theorem 4.20. Let Γ = Cay(Dic(A, y), S) be a Cayley graph over Dic(A, y) of order
4m. Then the splitting field of Γ is L
(√
γl | l ∈ C ′
)
, and the algebraic degree ofΓ satisfies
deg(Γ) =
φ(2m)|M |
|H ′|
,
where L = Q (ω2m)η
−1(H′)
= {x ∈ Q (ω2m) | σ(x) = x for all σ ∈ η−1(H ′)} and
γl, H
′, C ′,M are given in (4.16) – (4.19), respectively.
Corollary 4.21. Let Γ = Cay(Dic(A, y), S) be a Cayley graph over Dic(A, y) of order
4m. Then the algebraic degree of Γ satisfies
φ(2m)
|H ′|
≤ deg(Γ) ≤ φ(2m)2
|C′|
|H ′|
,
where H ′, C ′ are given in (4.17) and (4.18), respectively.
Corollary 4.22. Let Γ = Cay(Dic(A, y), S) be a Cayley graph over Dic(A, y) of order
4m. If H ′ = Z∗2m and γl is a square of an integer for each l ∈ C ′, where γl, H ′, C ′ are
given in (4.16) – (4.18), respectively, then Γ is integral.
Corollary 4.23. Let Γ = Cay(Dic(A, y), S) be a Cayley graph over Dic(A, y) of order
4m. Let
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈S
}
and T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
.
If T12 = T−112 , then the splitting field of Γ satisfies
SF(Γ) = Q (ω2m)η
−1(H′′)
= {x ∈ Q (ω2m) | σ(x) = x for all σ ∈ η−1(H ′′)},
the algebraic degree of Γ satisfies
deg(Γ) =
φ(2m)
|H ′′|
,
where H ′′ = {h ∈ Z∗2m | hT11 = T11, hT12 = T12}.
Corollary 4.24. Let Γ = Cay(Dic(A, y), S) be a Cayley graph over Dic(A, y) of order
4m. Let
T11 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ S
}
and T12 =
{
(t1, . . . , tr) |
(
at11 , . . . , a
tr
r
)
∈ xS
}
.
If T12 = T−112 and for all h ∈ Z∗2m, hT11 = T11 and hT12 = T12, then Γ is integral.
Furthermore, for the dicyclic group Dic4m =
〈
a, b | a2m = e, am = b2, b−1ab = a−1
〉
,
as direct consequences of Theorem 4.20 and Corollaries 4.21 – 4.24, we have similar re-
sults, so we omit the details here.
Example 4.25. Let Dic12 = ⟨a, b | a6 = e, a3 = b2, b−1ab = a−1⟩ be the dicyclic
group of order 12 and S =
{
a, a5, ab, a2b, a4b, a5b
}
. We consider the algebraic degree of
Γ = Cay(Dic12, S). Then T11 = {1, 5} and T12 = {1, 2, 4, 5} = T−112 . It follows that
H ′′ = {1, 5} = Z∗6. By Corollary 4.24, deg(Γ) = 1. In fact, Spec(Γ) = 4 ∗ {−1, 1} ∪ 3 ∗
{−2} ∪ {6}.
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 203
4.4 Cayley digraphs over semi-dihedral groups
For the semi-dihedral group SD8m, it has the following presentation
SD8m = ⟨a, b | a4m = b2 = e, bab = a2m−1⟩.
We now consider the algebraic degree of the Cayley digraph Γ = Cay(SD8m, S).
Using the symbols in Subsection 4.1, it follows that A = ⟨a⟩4m and A′ = Z4m. Let
x1 = e and x2 = b. Then
T11 =
{
t | at ∈ S
}
,
T12 =
{
t | bat ∈ S
}
,
T21 =
{
t | atb ∈ S
}
,
T22 =
{
t | a(2m−1)t ∈ S
}
.
In a similar way as in Subsection 4.1, we define
I1 = [t | t ∈ T11 or t ∈ T22] ,
I2 = [t | t ∈ (T11 + T11) or t ∈ (T22 + T22) or t ∈ 4 ∗ (T12 + T21)],
I3 = [t | t ∈ 2 ∗ (T11 + T22)].
Let
βl = χl(I1) and γl = χl(I2 \ I3)− χl(I3 \ I2), (4.20)
where χl(t) = ωlt4m and 0 ≤ l ≤ 4m − 1. Let K be a field such that Q ⊆ K ⊆ Q (ω4m).
Then Gal (Q (ω4m) /K)) ≤ Gal (Q (ω4m) /Q) ∼= Z∗4m. Let η : Gal (Q (ω4m) /Q) →
Z∗4m be the isomorphism such that σ (ω4m) = ω
η(σ)
4m , where σ ∈ Gal (Q (ω4m) /Q). Let
L = Q (βl, γl | 1 ≤ l ≤ 4m− 1) .
The following lemma helps to simplify the expression of L.
Lemma 4.26. If β1, γ1 ∈ K, then βl, γl ∈ K for 1 ≤ l ≤ 4m− 1.
Proof. The proof is similar to that of Lemma 4.15.
By Lemma 4.26, we have
L = Q (β1, γ1) .
Let
H ′ = {h ∈ Z∗4m | hI1 = I1, h(I2 \ I3) = I2 \ I3, h(I3 \ I2) = I3 \ I2} . (4.21)
By Proposition 4.3, we have H ′ = η (Gal (Q (ω4m) /L)).
Assume that H ′a(1), H ′a(2), . . . H ′a(k) are all distinct orbits of H ′ on Z4m. Let
C = {a(i) | Z4m ∩H ′a(i) ̸= ∅} (4.22)
and
M = ⟨[γl]L | l ∈ C⟩ . (4.23)
By Theorem 4.4 and Corollaries 4.5 – 4.8, we have
204 Ars Math. Contemp. 24 (2024) #P2.02 / 187–206
Theorem 4.27. Let Γ = Cay(SD8m, S) be a Cayley digraph over the semi-dihedral group
SD8m. Then the splitting field of Γ is L
(√
γl | l ∈ C
)
and the algebraic degree of Γ
satisfies
deg(Γ) =
φ(4m)|M |
|H ′|
,
where L = Q (ω4m)η
−1(H′)
= {x ∈ Q (ω4m) | σ(x) = x for all σ ∈ η−1(H ′)} and
γl, H
′, C,M are given in (4.20) – (4.23), respectively.
Corollary 4.28. Let Γ = Cay(SD8m, S) be a Cayley digraph over the semi-dihedral group
SD8m. Then the algebraic degree of Γ satisfies
φ(4m)
|H ′|
≤ deg(Γ) ≤ φ(4m)2
|C|
|H ′|
,
where H ′, C are given in (4.21) and (4.22), respectively.
Corollary 4.29. Let Γ = Cay(SD8m, S) be a Cayley digraph over the semi-dihedral group
SD8m. If H ′ = Z∗4m and γl is a square of an integer for each l ∈ C, where γl, H ′, C are
given in (4.20) – (4.22), respectively, then Γ is integral.
Corollary 4.30. Let Γ = Cay(SD8m, S) be a Cayley digraph over the semi-dihedral group
SD8m. If T11 = T22 and T12 = T−112 = T21, then the splitting field of Γ satisfies
SF(Γ) = Q (ω4m)η
−1(H′′)
= {x ∈ Q (ω4m) | σ(x) = x for all σ ∈ η−1(H ′′)},
the algebraic degree of Γ satisfies
deg(Γ) =
φ(4m)
|H ′′|
,
where H ′′ = {h ∈ Z∗4m | hT11 = T11, hT12 = T12}.
Corollary 4.31. Let Γ = Cay(SD8m, S) be a Cayley digraph over the semi-dihedral group
SD8m. If T11 = T22, T12 = T−112 = T21 and for all h ∈ Z∗4m, hT11 = T11 and hT12 = T12,
then Γ is integral.
We end this paper with the following example.
Example 4.32. Let SD16 = ⟨a, b | a8 = b2 = e, b−1ab = a3⟩ be the semi-dihedral
group of order 16 and S =
{
a2, a6, ba, ba5
}
. We consider the algebraic degree of Γ =
Cay(SD16, S). Then T11 = T22 = {2, 6} and T12 = {1, 5}, T−112 = T21 = {3, 7}. Thus,
I1 = 2 ∗ {2, 6}, I2 \ I3 = 8 ∗ {0, 4} and I3 \ I2 = ∅. It follows that H ′ = {1, 3, 5, 7} =
Z∗8. Since γ1 = 8[1 + (−1)l] is a square of integer for each l ∈ Z8, by Corollary 4.29,
deg(Γ) = 1. In fact, Spec(Γ) = 10 ∗ {0} ∪ 2 ∗ {−2, 1, 4}.
ORCID iDs
Lihua Feng https://orcid.org/0000-0003-4144-1649
Y. Wu et al.: Algebraic degrees of 2-Cayley digraphs over abelian groups 205
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.03 / 207–230
https://doi.org/10.26493/1855-3974.2894.b07
(Also available at http://amc-journal.eu)
A non-associative incidence near-ring with a
generalized Möbius function*
John Johnson †, Max Wakefield ‡
US Naval Academy, 572-C Holloway Rd, Annapolis MD, 21402 USA
This paper is dedicated to the memory of John Johnson.
Received 1 June 2022, accepted 27 February 2023, published online 20 September 2023
Abstract
There is a convolution product on 3-variable partial flag functions of a locally finite
poset that produces a generalized Möbius function. Under the product this generalized
Möbius function is a one sided inverse of the zeta function and satisfies many generaliza-
tions of classical results. In particular we prove analogues of Phillip Hall’s Theorem on
the Möbius function as an alternating sum of chain counts, Weisner’s Theorem, and Rota’s
Crosscut Theorem. A key ingredient to these results is that this function is an overlapping
product of classical Möbius functions. Using this generalized Möbius function we define
analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We
compute these polynomials for certain families of matroids and prove that this generalized
Möbius polynomial has -1 as root if the matroid is modular. Using results from Ardila and
Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.
Keywords: Incidence algebra, matroid, Möbius function, valuation.
Math. Subj. Class. (2020): 37K15, 42A99, 60E05, 05A17
*The authors are very thankful for detailed comments by the reviewer. The reviewers suggestions have signif-
icantly improved the article. The authors are thankful for discussions with Carolyn Chun, Joel Lewis, and Will
Traves. The authors are also thankful to George Andrews for help on Lemma 6.14. Frederico Ardila and Mario
Sanchez significantly helped with the material on valuations for which the authors are very thankful. Also, Jose
Bastidas made multiple excellent comments for which the authors are very thankful. The authors would like to
thank the US Naval Academy trident program for support during this project.
†Supported by the US Naval Academy as a Trident Scholar.
‡Corresponding author.
E-mail addresses: m213162@usna.edu (John Johnson), wakefiel@usna.edu (Max Wakefield)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
208 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
1 Introduction
Combinatorial invariants in incidence algebras play a central role in many areas of com-
binatorics as well as in number theory, algebraic topology, algebraic geometry, and repre-
sentation theory. In particular, the Möbius function appears in the inverse of the Riemann
zeta function as well as the coefficients of the chromatic polynomial for graphs. In this
note we study a generalization of the classical incidence algebra by looking at three vari-
able incidence functions. A large portion of this study is focussed on studying a 3-variable
generalized Möbius function inside this generalized incidence structure.
Incidence algebras and Möbius functions were popularized by Rota in [26]. Rota char-
acterized the classical Möbius function from number theory (see [20] and [14]) as the
inverse of the constant function 1 on the intervals of the poset which is called the zeta
function. In [26] Rota gives many results on the Möbius function, including his Crosscut
Theorem. Since then, many advances can be attributed to Möbius functions. Of particular
importance are the counting theorems of Zaslavsky in [33] and Terao’s factorization the-
orem (see [29]) using the Möbius function in the form of the characteristic polynomial of
a hyperplane arrangement. The main motivation for this work is to build invariants which
are finer than the classical Möbius function and characteristic polynomial to obtain more
information about the underlying combinatorial structure.
More recently, there has been considerable developments in understanding of some
classical invariants on matroids. One generalization came from Krajewski, Moffatt, and
Tanasa who built Tutte polynomials from a Hopf algebra in [18]. Taking this a little fur-
ther, in [11] Dupont, Fink and Moci construct a categorical framework to view various
combinatorial invariants and they prove some convolution formulas. The work of Aguiar
and Ardila in [1] framed many combinatorial structures like matroids in terms of general-
ized permutahedra, where there is a natural Hopf monoid governing classical operations.
One possible starting place for this study could be the work of Joni and Rota in [16]. Then,
in [6], Ardila and Sanchez use this Hopf monoid structure to build a concrete method for
investigating valuations on many combinatorial structures. Another aim of this study is to
add another invariant to the list of valuations. Concretely, we use the methods of Ardila and
Sanchez to show that one of our invariants is a valuation on matroids. One view that one
can take for many combinatorial structures is that of posets (e.g. matroids are geometric
lattices) and this is the view that we take here.
The starting point for our study is the collection of 3-variable functions on ordered
triples of elements in a poset. The set of these 3-variable functions also appears in the book
[2] by Aguiar and Mahajan in Appendix C4 where they study 2-cochains and 2-cocycles.
We differ from the work in Appendix C4 [2] by equipping this set of functions with a spe-
cial convolution product. The motivation for this product comes from trying to symmetrize
a more natural convolution product that was studied by the second author in [30] as well as
making new invariants with special properties. This product provides a 3-variable Möbius
function which is a sort of left inverse of the 3-variable analogue of the zeta function. We
call this function the J-function and study many of its properties. It turns out that it is
essentially a staggered product of the classical Möbius functions and hence satisfies gen-
eralizations of many of the classical theorems on the classical Möbius function. To prove
these results we develop and use certain operations and formulas these 3-variable func-
tions satisfy that give maps between various different types of incidence algebras. In [8]
Jose Bastidas studies Type B Hopf monoids and defines an antipode via some convolution
formulas which seem to have some similar properties to the work presented here.
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 209
As an application we build two different polynomials from the J-function: a general-
ized characteristic polynomial and a generalized Möbius polynomial (see [17] and [21] for
Möbius polynomials). It turns out that these polynomials have some interesting properties
that are not apparent from the surface. In the case of matroids, the generalized character-
istic polynomial has positive coefficients. We then compute these polynomials for certain
families of matroids and find special roots. Of particular interest is that the generalized
Möbius function has −1 as a root for modular matroids, which mimics Theorem 1 in [21].
However, we show that the converse is not true and so one is led to question what do these
polynomial count? Could there be some chromatic generalization for the generalized poly-
nomials, or some lattice point or finite field counting formula for these polynomials (like
[9] or [7])? Also, in [8], Bastidas defines some polynomial invariants via characters of a
Hopf monoid. Can the polynomials we define here be put in the framework of [8]?
We finish by employing the methods of Ardila and Sanchez in [6] to show that our
generalized characteristic polynomial is a matroid valuation. This follows from the fact
that the J-function splits as a product of Möbius functions. In the case of the Möbius
polynomial, we are not sure whether or not it is a valuation, yet we show that it does have
a decomposition in terms of the classical characteristic polynomials. We find it interest-
ing that this decomposition looks very similar to the recursive definition of the matroid
Kazhdan-Lusztig polynomial originally defined in [12].
We begin this study with reviewing classical results on incidence algebras and Möbius
functions in Section 2. Then we define our 3-variable incidence structure in Section 3.
There we show that this structure has some interesting properties but that it is neither as-
sociative nor distributive. However, in Section 4 we develop multiple operations which
give nice formulas between these different kinds of incidence functions. Using these for-
mulas we define a generalized Möbius function, the J-function, and study its properties
in Section 5. Finally in Section 6 we define our generalized characteristic and Möbius
polynomials.
2 Incidence Algebras
Let R be a commutative ring and P be a locally finite poset. We follow [28] and [4] for
combinatorics on posets. For the remainder of this note we refer to the order in P by ≤.
Also, for n ∈ N let [n] = {1, 2, 3, . . . , n}. In this section we review basic material of
incidence algebras where we follow [27]. First we define the poset of partial flags.
Definition 2.1. The poset of partial flags of length k on P is
F lk(P) =
{
(x1, x2, . . . , xk) ∈ Pk| x1 ≤ x2 ≤ · · · ≤ xk
}
with order given by (x1, . . . , xk) ⪯ (y1, . . . , yk) if and only if for all i ∈ [k] we have
xi ≤ yi.
Now we define the classical incidence algebras.
Definition 2.2. The incidence algebra on P is the set
I(P, R) = Hom(F l2(P), R)
where R is a commutative ring. Addition in I(P, R) is given by
(f + g)(x, y) = f(x, y) + g(x, y),
210 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
the multiplication is given by convolution
(f ∗ g)(x, y) =
∑
x≤a≤b
f(x, a)g(a, y),
and the scalar product is given by (rf)(x, y) = rf(x, y) for all r ∈ R.
In this note, we will examine multiple different operations on functions on posets. For
this reason we will reserve juxtaposition only for products of elements in the ring R. Oth-
erwise we will denote products of functions with specific operation names like ∗.
It turns out that I(P, R) is a non-commutative R-algebra with identity element given
by the Kronecker delta function
δ(x, y) =
{
1 if x = y,
0 else.
There are two other very important elements in I(P, R).
Definition 2.3. The zeta function ζ ∈ I(P, R) is defined as the constant function on
F l2(P)
ζ(x, y) = 1
for all (x, y) ∈ F l2(P). The Möbius function µ ∈ I(P, R) is defined by∑
x≤a≤y
µ(x, a) =
∑
x≤a≤y
µ(a, y) = δ(x, y)
for all (x, y) ∈ F l2(P).
The Möbius function was originally defined by Möbius (see [20]) on the poset of the
natural numbers ordered by division for the purpose of inverting the Riemann zeta function.
Since then the Möbius function has been used in many different contexts and broadened by
the work of Rota in [26]. For our discussion, it is important to note that µ is the multiplica-
tive inverse of the zeta function
µ ∗ ζ = ζ ∗ µ = δ.
Now we review how the incidence algebra functor factors over products. Recall that for
posets P and Q the product poset is P ×Q with order given by (x1, x2) ≤ (y1, y2) if and
only if x1 ≤ y1 and x2 ≤ y2.
Proposition 2.4 (Proposition 2.1.12 [27]). If P and Q are locally finite posets then
I(P, R)⊗R I(Q, R) ∼= I(P ×Q, R).
Because of Proposition 2.4 we define the following operation on functions. In order to
the make the exposition clear in the case when we are dealing with functions over different
posets, we will put the poset in the subscript. For fP ∈ I(P, R) and gQ ∈ I(Q, R) define
fP × gQ ∈ I(P ×Q, R) by
(fP × gQ)((x1, x2), (y1, y2)) = fP(x1, y1)gQ(x2, y2).
We will use this notation and the following consequence of Proposition 2.4 in our study in
Section 5.
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 211
Corollary 2.5. If P and Q are locally finite posets then µP × µQ = µP×Q.
Next we recall how the Möbius function counts chains (or is an Euler characteristic for
the order complex). For (x, y) ∈ F l2(P) let
ci(x, y) =
∣∣{(a0, . . . , ai) ∈ F li+1 : ∀k, ak < ak+1 and a0 = x and ai = y}∣∣
be the number of chains of length i between x and y.
Theorem 2.6 (Phillip Hall’s Theorem [13]; Proposition 3.8.5 [28]). If P is a locally finite
poset and (x, y) ∈ F l2(P) then
µ(x, y) =
∑
i
(−1)ici(x, y).
Now we review Rota’s Crosscut Theorem. Let L be a finite lattice with 0̂ the minimum
element and 1̂ the maximum element. Usually, Rota’s Crosscut Theorem is stated globally
in the lattice giving a formula for µ(0̂, 1̂). However, for our generalization we will need a
local version.
Definition 2.7. Let (x, y) ∈ F l2(L). A lower crosscut of the interval [x, y] = {a ∈ L|x ≤
a ≤ y} is a set Sx,y ⊆ [x, y]\{x} such that if b ∈ [x, y]\(Sx,y ∪ {x}) then there is some
a ∈ Sx,y with a < b. A upper crosscut of the interval [x, y] is a set Tx,y ⊆ [x, y]\{y} such
that if a ∈ [x, y]\(Tx,y ∪ {y}) then there is some b ∈ Tx,y with a < b.
This definition gives Rota’s famous Crosscut Theorem which we state in the style of
Lemma 2.35 in [22] for use in arrangement theory.
Theorem 2.8 ([26, Theorem 3]). If L is a lattice, (x, y) ∈ F l2(L), and Sx,y is a lower
crosscut of [x, y] then
µ(x, y) =
∑
A⊆Sx,y∨
A=y
(−1)|A|.
Dually, if Tx,y is an upper crosscut of [x, y] then
µ(x, y) =
∑
B⊆Tx,y∧
B=x
(−1)|B|.
Next we consider Weisner’s Theorem (see [31]).
Theorem 2.9 ([28, Weisner’s Theorem, Corollary 3.9.3]). If L is a finite lattice with at
least two elements and 1̂ ̸= a ∈ L then∑
x∈L
x∧a=0̂
µ(x, 1̂) = 0.
Now we recall one more result that follows from this classical result for matroids: the
Mobius function of the lattice of flats of a matroid alternates in sign.
Lemma 2.10. If L is a finite semimodular lattice then sgn(µ(x, y)) = (−1)rk(x)+rk(y).
212 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
3 A 3-variable incidence non-associative near-ring
In this section, we define the algebraic structures where our invariants live. It turns out that
these algebraic structures support various operations that can yield nice formulas. Later
these formulas will be used to show certain formulas and relations on our new invariants.
Definition 3.1. Let R be a commutative ring and P be a locally finite poset. Define the
3-variable incidence left near-ring as
J(P, R) = Hom(F l3(P),R)
with binary operations as follows:
• For f, g ∈ J(P, R) we define addition by
(f + g)(x, y, z) = f(x, y, z) + g(x, y, z).
• For f, g ∈ J (P, R) we define a multiplication by
(f  g)(x, y, z) =
∑
(a,b)⊴(x,y,z)
f(x, a, a)g(a, y, b)f(b, b, z)
where the juxtaposition in each term is multiplication in the ring R and (a, b) ⊴
(x, y, z) means x ≤ a ≤ y ≤ b ≤ z in P .
First we show that J(P, R) is indeed left distributive.
Proposition 3.2. If P is any poset then the multiplication  in J(P, R) is left distributive.
Proof. Let f, g, h ∈ J(P, R) and (x, y, z) ∈ F l3(P). Then
(f  (g + h))(x, y, z) =
∑
(a,b)⊴(x,y,z)
f(x, a, a)(g + h)(a, y, b)f(b, b, z)
=
∑
(a,b)⊴(x,y,z)
f(x, a, a)(g(a, y, b) + h(a, y, b))f(b, b, z)
=
∑
(a,b)⊴(x,y,z)
f(x, a, a)g(a, y, b)f(b, b, z)
+
∑
(a,b)⊴(x,y,z)
f(x, a, a)h(a, y, b)f(b, b, z)
= (f  g)(x, y, z) + (f  h)(x, y, z).
Remark 3.3. With this + the set J(P, R) is an abelian group. It would be convenient if
J(P, R) were naturally an R-algebra. However, this is far from the case as we will see.
Even the natural action of R on J(P, R) is flawed. Let r ∈ R and f, g ∈ J(P, R) then
r · (f  g) = f  (r · g) but (r · f)  g = r2 · (f  g).
Fortunately, though, there are a few special functions in J(P, R) that provide substantial
information. We will use these to study the structure of J(P, R) and define other special
elements later.
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 213
Definition 3.4. Assume that 1 is the multiplicative identity and 0 is the additive identity
in R.
• Define δ3 ∈ J(P, R) by
δ3(x, y, z) =
{
1 if x = y = z
0 otherwise
• Define ζ3 ∈ J(P, R) by setting ζ3(x, y, z) = 1 for all (x, y, z) ∈ F l3(P).
With these functions we can investigate basic properties of J(P, R).
Proposition 3.5. The element δ3 ∈ J(P, R) is a left multiplicative identity.
Proof. Let f ∈ J(P, R) and (x, y, z) ∈ F l3(P). Then
(δ3  f)(x, y, z) =
∑
(a,b)⊴(x,y,z)
δ3(x, a, a)f(a, y, b)δ3(b, b, z)
= δ3(x, x, x)f(x, y, z)δ3(z, z, z) = f(x, y, z).
In the next three propositions we note that in general J(P, R) is not commutative, as-
sociative, or right distributive. We could do this with a single example, however these
propositions show that J(P, R) is basically never commutative, associative, or right dis-
tributive.
Proposition 3.6. If P is a non-trivial poset (it has at least two comparable elements) or
the base ring is not Boolean (not idempotent), then the multiplication  in J(P, R) is non-
commutative and δ3 is not a right multiplicative identity.
Proof. Let (x, y, z) ∈ F l3(P) and suppose that either x < y or that y < z in P or that R
is not Boolean. Under these assumptions we can construct a function f ∈ J(P, R) that has
f(x, y, z) ̸= f(x, y, y)f(y, y, z). Then from Proposition 3.5 we have (δ3  f)(x, y, z) =
f(x, y, z) but (f  δ3)(x, y, z) = f(x, y, y)f(y, y, z).
The proof for the next fact is very similar.
Proposition 3.7. If P is a poset with three elements x, y, z satisfying x < y < z or the
base ring is not Boolean (not idempotent), then the multiplication  in J(P, R) is non-
associative.
Proof. Let (x, y, z) ∈ F l3(P) be three elements satisfying x < y < z in P or that R is
not Boolean. Under these assumptions we can construct a function f ∈ J(P, R) that has
f(x, y, y)f(y, y, y)2f(y, y, z) ̸= f(x, y, y)f(y, y, z). Compute
((f  δ3)  δ3))(x, y, z) =
∑
(a,b)⊴(x,y,z)
(f  δ3)(x, a, a)δ3(a, y, b)(f  δ3)(b, b, z)
= [(f  δ3)(x, y, y)][(f  δ3)(y, y, z)]
= [f(x, y, y)f(y, y, y)][f(y, y, y)f(y, y, z)].
Then from Proposition 3.5 we have (f  (δ3  δ3))(x, y, z) = (f  δ3)(x, y, z) =
f(x, y, y)f(y, y, z) which is different from ((f  δ3)  δ3))(x, y, z) by our assumption
on f .
214 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Proposition 3.8. If P is a non-trivial poset (it has at least two comparable elements) and
R is any non-trivial commutative ring, then the multiplication  in J(P, R) is not right
distributive.
Proof. Let (x, y, z) ∈ F l3(P) and f ∈ J(P, R) be any function such that f(x, y, y) +
f(y, y, z) ̸= 0. Then
((f + ζ3)  δ3)(x, y, z) =
∑
(a,b)⊴(x,y,z)
(f + ζ3)(x, a, a)δ3(a, y, b)(f + ζ3)(b, b, z)
= [(f + ζ3)(x, y, y)][(f + ζ3)(y, y, z)]
= f(x, y, y)f(y, y, z) + f(x, y, y) + f(y, y, z) + 1.
On the other hand we have
((f  δ3) + (ζ3  δ3))(x, y, z) = f(x, y, y)f(y, y, z) + ζ3(x, y, y)ζ3(y, y, z)
= f(x, y, y)f(y, y, z) + 1
which by the hypothesis on f we have the right distributive property not holding.
With Propositions 3.5, 3.6, 3.7, 3.2, and 3.8 we conclude that J(P, R) is a left only
unital, non-commutative, non-associative, near-ring (see [25] for this terminology). Also,
note that there is the zero function Z ∈ J(P, R) which satisfies Z  f = f  Z = Z for
all f ∈ J(P, R). Further note that addition in J(P, R) is abelian. Hence J(P, R) is an
abelian, zero-symmetric, left only unital, non-commutative, non-associative, near-ring. It
is worth noting that in general J(P, R) is not even close to being associative on both sides
and is not an alternative algebra or any similar generalization.
Now we look at a few special cases that do not satisfy the hypothesis of some of these
propositions.
Example 3.9. Let P = B0 = {0} be the poset with just one element and R any commu-
tative ring. Then as a set J(B0, R) = R, but multiplication is given by a  b = aba = a2b.
If R is Boolean then J(B0, R) ∼= R. Otherwise, this near-ring is not associative, not com-
mutative, and is only left unital.
Example 3.10. Let P = B1 = {0, 1} be the Boolean poset of rank 1 and R be any Boolean
ring (one example would be F2). Then the hypothesis of Proposition 3.7 is not satisfied and
the non-equality f(x, y, y)f(y, y, y)2f(y, y, z) ̸= f(x, y, y)f(y, y, z) used in the proof is
always equal. It turns out that in this case J(B1, R) is associative and we prove this now.
In order to shorten the calculation we will denote (0, 0, 0) by 0⃗ and (1, 1, 1) by 1⃗. First we
see that
((f  g)  h)(⃗0) = f (⃗0)g(⃗0)h(⃗0) = (f  (g  h))(⃗0).
Then for the non-trivial tuple (0, 0, 1) we compute
((f  g)  h)(0, 0, 1) =(f  g)(⃗0)h(⃗0)(f  g)(0, 0, 1)
+ (f  g)(⃗0)h(0, 0, 1)(f  g)(⃗1)
=f (⃗0)g(⃗0)h(⃗0)[f (⃗0)g(⃗0)f(0, 0, 1) + f (⃗0)g(0, 0, 1)f (⃗1)]
+ f (⃗0)g(⃗0)h(0, 0, 1)f (⃗1)g(⃗1)
=f (⃗0)g(⃗0)h(⃗0)f(0, 0, 1) + f (⃗0)g(⃗0)h(⃗0)g(0, 0, 1)f (⃗1)
+ f (⃗0)g(⃗0)h(0, 0, 1)f (⃗1)g(⃗1).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 215
Then the other side of the associative identity is
(f  (g  h))(0, 0, 1) =f (⃗0)(g  h)(⃗0)f(0, 0, 1) + f (⃗0)
[
(g  h)(0, 0, 1)
]
f (⃗1)
=f (⃗0)g(⃗0)h(⃗0)f(0, 0, 1) + f (⃗0)
[
g(⃗0)h(⃗0)g(0, 0, 1)
+ g(⃗0)h(0, 0, 1)g(⃗1)
]
f (⃗1)
=((f  g)  h)(0, 0, 1).
Hence J(B1, R) is associative. This example does satisfy the hypothesis of Proposition 3.8.
Hence J(B1, R) is a (associative) left abelian (addition is commutative) near-ring. That’s
about as good as it gets though. For example, if R = F2 then J(B1,F2) is not a near-field
because any function with f (⃗0) = 0 and f(0, 0, 1) = 1 does not have an inverse. For
exactly the same reason δ3 ∈ J(B1,F2) is still not a right identity element.
4 Operations on incidence functions
In this section we look at a relationship between the classical incidence algebra I(P, R)
and J(P, R). For f, g ∈ I(P, R) we define f♢g ∈ J(P, R) by setting
(f♢g)(x, y, z) = f(x, y)g(y, z).
We can use the ♢ operation to construct interesting elements in J(P, R). There are rela-
tionships between the operations ∗ in I(P, R),  in J(P, R), and ♢.
Proposition 4.1. If f, g, r, s ∈ I(P, R) and f(b, b)g(a, a) = 1 for all a, b ∈ P then
(f♢g)  (r♢s) = (f ∗ r)♢(s ∗ g).
Proof. Let (x, y, z) ∈ F l3(P) and f, g, r, s ∈ I(P, R). Then
((f♢g)  (r♢s))(x, y, z) =
∑
(a,b)⊴(x,y,z)
(f♢g)(x, a, a)(r♢s)(a, y, b)(f♢g)(b, b, z)
=
∑
(a,b)⊴(x,y,z)
f(x, a)g(a, a)r(a, y)s(y, b)f(b, b)g(b, z)
=
 ∑
x≤a≤y
f(x, a)r(a, y)
 ∑
y≤b≤z
s(y, b)g(b, z)

= [(f ∗ r)(x, y)] [(s ∗ g)(y, z)]
=((f ∗ r)♢(s ∗ g))(x, y, z)
where the third equality only holds due the the assumption.
One can see from the proof that without the hypothesis on f and g that the equality will
not hold. Hence there is no hope for this to give any kind of near-ring homomorphism from
a twisted product version of I(P, R)× I(P, R). Also, the natural addition homomorphism
assumption does not hold. Instead we have the following proposition which does not have
special hypothesis on the functions. For this proposition there are two different additions,
for I(P, R) and J(P, R), which for brevity we use the same addition symbol.
216 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Proposition 4.2. If f, g, r, s ∈ I(P, R) then
(f + g)♢(r + s) = (f♢r) + (f♢s) + (g♢r) + (g♢s).
Proof. For all (x, y, z) ∈ F l3(P)
((f + g)♢(r + s))(x, y, z) =(f(x, y) + g(x, y))(r(y, z) + s(y, z))
=f(x, y)r(y, z) + f(x, y)s(y, z) + g(x, y)r(y, z)+
g(x, y)s(y, z)
=((f♢r) + (f♢s) + (g♢r) + (g♢s))(x, y, z)
which is the identity we are looking for.
We can also define products of functions on products of posets over 3-flags. We prefer
to limit our study of J(P, R) to this product definition since the technicalities of tensor
products over non-associative near-rings would present significant and unnecessary com-
plications.
Definition 4.3. Let P and Q be locally finite posets, fP ∈ J(P, R), and gQ ∈ J(Q, R).
Define fP × gQ ∈ J(P ×Q, R) by
(fP × gQ)((x1, x2), (y1, y2), (z1, z2)) = fP(x1, y1, z1)gQ(x2, y2, z2).
Now we show how the ♢ operation is compatible with products of posets.
Proposition 4.4. If P and Q are locally finite posets, fP , gP ∈ I(P, R), and rQ, sQ ∈
I(Q, R) then
(fP♢gP)× (rQ♢sQ) = (fP × rQ)♢(gP × sQ).
Proof. Let ((x1, x2), (y1, y2), (z1, z2)) ∈ F l3(P ×Q). Then
((f♢g)× (r♢s))((x1, x2), (y1, y2), (z1, z2))
= [(f♢g)(x1, y1, z1)] [(r♢s)(x2, y2, z2)]
= [f(x1, y1)g(y1, z1)] [r(x2, y2)s(y2, z2)]
= [f(x1, y1)r(x2, y2)] [g(y1, z1)s(y2, z2)]
= [(f × r)((x1, y1), (x2, y2))] [(g × s)((y1, z1), (y2, z2))]
= ((f × r)♢(g × s))((x1, x2), (y1, y2), (z1, z2))
which completes the proof.
As in Proposition 4.4 we will now show how the operations × and  factor over prod-
ucts of posets. We use subscripts on these operations to keep track of which poset the
operation is applied.
Proposition 4.5. If P and Q be locally finite posets, fP , gP ∈ J(P, R), and rQ, sQ ∈
J(Q, R) then
(fP P gP)× (rQ Q sQ) = (fP × rQ) P×Q (gP × sQ).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 217
Proof. Let x = (x1, x2), y = (y1, y2), z = (z1, z2) ∈ P ×Q so that (x, y, z) ∈ F l3(P ×
Q) and a = (a1, a2), b = (b1, b2) ∈ P ×Q so that (a, b) ∈ F l2(P ×Q). Then
((fP×rQ) P×Q (gP × sQ))(x, y, z)
=
∑
(a,b)⊴(x,y,z)
(fP × rQ)(x, a, a)(gP × sQ)(a, y, b)(fP × rQ)(b, b, z)
=
∑
(a1,b1)
∑
(a2,b2)
fP(x1, a1, a1)gP(a1, y1, b1)fP(b1, b1, z1)
rQ(x2, a2, a2)sQ(a2, y2, b2)rQ(b2, b2, z2)
= [(fP  gP)(x1, y1, z1)] [(rQ  sQ)(x2, y2, z2)]
= ((fP P gP)× (rQ Q sQ))(x, y, z)
which is the required identity.
5 The J-function
Let P be a locally finite poset. In this section we define the central invariant of this note
which we call the J function. This function is a generalization of the classical Möbius
function µ. We show that it satisfies generalizations of the classical theorems on µ. A key
ingredient for these results is the operation ♢.
Definition 5.1. Define J : F l3(P) → Z for all fixed (x, y, z) ∈ F l3(P ) by∑
(a,b)⊴(x,y,z)
J(a, y, b) = δ3(x, y, z).
This function is well defined because either x = y = z with J(x, y, z) = 1 or otherwise
all of the following summations are finite
J(x, y, z) =−
∑
x<a<y
 ∑
y<b<z
J(a, y, b)

−
∑
x<a≤y
J(a, y, z)−
∑
y≤b<z
J(x, y, b).
Note that J is exactly the function in J(P, R) such that
ζ3  J = δ3. (5.1)
This is a good reason why we say it is a generalization of the classical Möbius function
and below we show that there are a few more interesting reasons. It turns out that this
function was actually defined before in [30] with the notation µp3 and is exactly given by
the ♢ product construction in the previous section.
Theorem 5.2. For any locally finite poset P we have J = µp3 = µ♢µ.
218 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Proof. This follows from Proposition 4.1 since ζ ∈ I(P, R) satisfies the hypothesis and
ζ3  (µ♢µ) = (ζ♢ζ)  (µ♢µ) = (ζ ∗ µ)♢(µ ∗ ζ) = δ♢δ = δ3.
Hence J and µ♢µ satisfy the same recursive definition.
Now we can use all the classical properties of µ to conclude information about J . We
start by noticing that J is also a left inverse of ζ3.
Corollary 5.3. J  ζ3 = δ3.
Proof. Since µ satisfies the hypothesis of Proposition 4.1 we get
J  ζ3 = (µ♢µ)  (ζ♢ζ) = (µ ∗ ζ)♢(ζ ∗ µ) = δ♢δ = δ3
which is the desired result.
Interpreting Corollary 5.3 in terms of the definition and sums in the ring R we get the
following.
Corollary 5.4. For any locally finite poset P and (x, y, z) ∈ F l3(P) we have∑
(a,b)⊴(x,y,z)
J(x, a, a)J(b, b, z) = δ3(x, y, z)
and in particular ∑
(a,b)⊴(x,y,z)
µ(x, a)µ(b, z) = δ3(x, y, z).
Now we look at how the J function behaves over products. It turns out that J factors
over products.
Proposition 5.5. If P and Q are locally finite posets then JP × JQ = JP×Q.
Proof. For posets P and Q we have JP × JQ = (µP♢µP)× (µQ♢µQ) by definition. By
Proposition 4.4 (µP♢µP) × (µQ♢µQ) = (µP × µQ)♢(µP × µQ). Then using Proposi-
tion 2.5 we get (µP × µQ)♢(µP × µQ) = µP×Q♢µP×Q = JP×Q.
Next we look at a generalization of Phillip Hall’s Theorem. For (x, y, z) ∈ F l3(P) set
ci,j(x, y, z)=
∣∣{(a0, . . . , ai+j)∈F li+j+1 :∀k, ak<ak+1 and a0 = x, ai = y, ai+j = z}∣∣.
There is a bijection between the underlying set of ci,j(x, y, z) to the product of the under
lying sets of ci(x, y) and cj(y, z). This results in the following.
Lemma 5.6. If P is a locally finite poset and (x, y, z) ∈ F l3(P) then ci,j(x, y, z) =
ci(x, y)cj(y, z).
This leads to a generalization of Phillip Hall’s Theorem for the J function.
Theorem 5.7. If P is a locally finite poset and (x, y, z) ∈ F l3(P) then
J(x, y, z) =
∑
i,j∈N
(−1)i+jci,j(x, y, z).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 219
Proof. Let (x, y, z) ∈ F l3(P). By Theorem 5.2 J(x, y, z) = µ(x, y)µ(y, z). Then using
Theorem 2.6 we get
J(x, y, z) =
[∑
i∈N
(−1)ici(x, y)
]∑
j∈N
(−1)jcj(y, z)

=
∑
i,j∈N
(−1)i+jci(x, y)ci(y, z).
Lemma 5.6 finishes the proof.
Now we focus on a version of Rota’s Crosscut Theorem for the J function. We state
this following the style of Lemma 2.35 in [22] and Theorem 2.4.9 in [19], which are forms
of Rota’s original Crosscut Theorem in [26]. To state this result we need the following
definition.
Definition 5.8. Let L be a finite lattice, (x, y, z) ∈ F l3(L), Sx,y be a lower crosscut
of [x, y], and Sy,z be a lower crosscut of [y, z] as in Definition 2.7. We call Sx,y,z =
Sx,y
⊔
Sy,z a double lower crosscut of (x, y, z) and call Sx,y and Sy,z the components of
Sx,y,z . Similarly we can define Tx,y,z = Tx,y
⊔
Ty,z (as well as STx,y,z = Sx,y
⊔
Ty,z
and TSx,y,z = Tx,y
⊔
Sy,z).
Theorem 5.9. If L is a finite lattice, (x, y, z) ∈ F l3(L), Sx,y,z is a double lower crosscut
of (x, y, z) with components Sx,y and Sy,z then
J(x, y, z) =
∑
A⊆Sx,y,z∨
(A∩Sx,y)=y∨
(A∩Sy,z)=z
(−1)|A|.
Proof. Again we use Theorem 5.2 together with the classical Theorem 2.8
J(x, y, z) =µ(x, y)µ(y, z)
=
 ∑
A1⊆Sx,y∨
A1=y
(−1)|A1|

 ∑
A2⊆Sy,z∨
A2=z
(−1)|A2|

=
∑
A1⊆Sx,y∨
A1=y
∑
A2⊆Sy,z∨
A2=z
(−1)|A1|+|A2|.
Since the union in Definition 5.8 is disjoint |A1|+ |A2| = |A1
⊔
A2| and we have finished
the proof.
We end this section with a generalization of Weisner’s Theorem 2.9. The interesting
observation of this fact is that the middle variable of the function is crucial.
Theorem 5.10. If L is a finite lattice with at least three elements and 0̂ < a < b ∈ L then∑
x∈L
x∧a=0̂
J(x, b, 1̂) = 0.
220 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Proof. We compute the sum again using Theorem 5.2:∑
x∈L
x∧a=0̂
J(x, b, 1̂) =
∑
x
x∧a=0̂
µ(x, b)µ(b, 1̂)
= µ(b, 1̂)
∑
x
x∧a=0̂
µ(x, b)
= µ(b, 1̂) · 0 = 0
since a < b we can apply Weisner’s Theorem 2.9.
Remark 5.11. There is a dual version of this result where we sum over the left most
variable as in [26]. However, we do not see a version that sums over the middle variable.
6 Generalized characteristic and Möbius polynomials
In this section we examine two polynomials defined by summing over all values of the J
function on a ranked poset. One mimics the characteristic polynomial of a matroid and the
other looks like a one variable Möbius polynomial. We find more interesting information
inside the generalized Möbius polynomial than the generalized characteristic polynomial.
That is opposite of the state of affairs in the literature on the classical polynomials, but we
do not know why.
Definition 6.1. For P a ranked finite poset with minimum element 0̂ and maximum element
1̂ the J-characteristic polynomial of P is
J (P, t) = (−1)rk(P)
∑
x∈P
J(0̂, x, 1̂)trk(P)−rk(x).
Definition 6.2. Let P be a ranked finite poset and for (x, y, z) ∈ F l3(P) let ρ(x, y, z) =
3rk(P)− rk(x)− rk(y)− rk(z). The J-Möbius polynomial of P is
M(P, t) =
∑
(x,y,z)∈Fl3(P)
J(x, y, z)tρ(x,y,z).
We may sometimes refer to rk(P)− rk(x) as crk(x). These polynomials satisfy some
nice basic properties. For example it turns out that the coefficients of J (P, t) are positive
for nice P . For convenience if L is a ranked poset let Lk = {x ∈ L|rk(x) = k}.
Proposition 6.3. If L is a finite semimodular lattice then the coefficients of J (L, t) are
positive.
Proof. Using Theorem 5.2 we get that
J (L, t) = (−1)rk(L)
∑
x∈L
µ(0̂, x)µ(x, 1̂)trk(L)−rk(x).
So, the coefficient of tk is
ck = (−1)rk(L)
∑
x∈Lk
µ(0̂, x)µ(x, 1̂).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 221
Then note that by applying Lemma 2.10 we have
sgn(µ(0̂, x)µ(x, 1̂)) = (−1)rk(0̂)+rk(x)(−1)rk(x)+rk(1̂) = (−1)rk(L).
Hence sgn(ck) = (−1)2rk(L) = 1.
Now we look at a foundational property for the J-Möbius polynomial.
Proposition 6.4. If L is a finite lattice with at least two elements then M(L, 1) = 0.
Proof. Since L is a finite lattice with at least two elements we know there is a minimum
element 0̂ and a maximum element 1̂. Then
M(P, 1) =
∑
(x,y,z)∈Fl3(L)
J(x, y, z)
=
∑
y∈L
 ∑
(x,z)⊴(0̂,y,1̂)
J(x, y, z)

=
∑
y∈L
[
δ3(0̂, y, 1̂)
]
.
Since L has at least two elements 0̂ ̸= 1̂ so δ3(0̂, y, 1̂) is zero for all y.
We also have products formulas for both of these polynomials.
Proposition 6.5. If P and Q are ranked finite posets then J (P×Q, t) = J (P, t)J (Q, t).
Proof. Using Proposition 5.5 we get that
J (P, t)J (Q, t) =
(−1)rk(P)∑
p∈P
JP(0̂, p, 1̂)t
crk(p)
(−1)rk(Q) ∑
q∈Q
JQ(0̂, q, 1̂)t
crk(q)

= (−1)rk(P)+rk(Q)
∑
p∈P
∑
q∈Q
JP(0̂, p, 1̂)JQ(0̂, q, 1̂)t
crk(p)+crk(q)
= (−1)rk(P×Q)
∑
(p,q)∈P×Q
JP×Q((0̂, 0̂), (p, q), (1̂, 1̂))t
crk(p,q)
= J (P ×Q, t).
The proof of the following is almost identical.
Proposition 6.6. If P and Q are ranked finite posets then M(P × Q, t) =
M(P, t)M(Q, t).
Now we can use these product formulas to establish formulas for Boolean matroids.
Proposition 6.7. If Bn is the Boolean lattice then
J (Bn, t) = (t+ 1)n.
222 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Proof. We start with B1. This poset has two elements B1 = {0, 1}. So, J (B1, t) =
(−1)(J(0, 0, 1)t1 + J(0, 1, 1)t0) = t+ 1. Then the result follows since Bn = (B1)n.
Proposition 6.8. If Bn is the Boolean lattice then
M(Bn, t) = (t+ 1)n(t− 1)2n.
Proof. Again we first compute M(B1, t). The only coefficients are J(0, 0, 0) = 1,
J(0, 0, 1) = −1, J(0, 1, 1) = −1 and J(1, 1, 1) = 1. Then the result follows from
M(B1, t) =J(0, 0, 0)t3 + J(0, 0, 1)t2 + J(0, 1, 1)t+ J(1, 1, 1)
=t3 − t2 − t+ 1
=(t+ 1)(t− 1)2
and the application of Proposition 6.6.
Proposition 6.9. Let Pn be a geometric lattice of rank two with n atoms (rank 2 matroid
with n elements a.k.a. U2,n). Then M(Pn, t) = (t2 − nt+ 1)(t+ 1)2(t− 1)2.
Proof. We prove this by induction on n. The base case is n = 2 and is given by the n = 2
version of Proposition 6.8. Now assume n > 2. The lattice Pn consists of 0̂, 1̂, and n
atoms α1, . . . , αn. Now JPn(0̂, 0̂, 1̂) = n − 1 and JPn(0̂, 1̂, 1̂) = n − 1 are the only JPn
values that do not have αn as an entry and incorporate αn in it’s recursive definition. So,
JPn(0̂, 0̂, 1̂) = JPn−1(0̂, 0̂, 1̂) + 1 and similarly for (0̂, 1̂, 1̂). Incorporating this difference
into the calculation we get that
M(Pn, t) =M(Pn−1, t) + t4 + t2 + J(0̂, 0̂, αn)t5 + J(0̂, αn, αn)t4 + J(0̂, αn, 1̂)t3
+ J(αn, αn, αn)t
3 + J(αn, αn, 1̂)t
2 + J(αn, 1̂, 1̂)t
=(t2 − (n− 1)t+ 1)(t+ 1)2(t− 1)2 − (t5 − 2t3 + t)
=(t2 − nt+ 1)(t+ 1)2(t− 1)2
which is the desired formula.
Now we consider a decomposition of M(L, t) for a finite lattice L. If L is a finite
lattice then Lop is the same underlying set as L but with the order reversed (i.e. x ≤op y
in Lop if and only if x ≥ y in L). Also for y ∈ L let Ly = {x ∈ L|x ≤ y} and
Ly = {x ∈ L|x ≥ y}. Now we can state the result.
Proposition 6.10. If L is a finite ranked lattice then
M(L, t) = trk(L)
∑
y∈L
tcrk(y)χ(Ly, t)χ((Lop)y, t−1).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 223
Proof. First we note that for x ≤ y ∈ L the Möbius function on Lop has µop(y, x) =
µ(x, y) and that rank is corank in Lop. Then again using Theorem 5.2 we compute
M(L, t) =
∑
(x,y,z)∈Fl3(P)
J(x, y, z)tρ(x,y,z)
=
∑
y∈L
∑
x≤y
∑
z≥y
µ(x, y)µ(y, z)tcrk(x)+crk(y)+crk(z)
=
∑
y∈L
tcrk(y)
∑
x≤y
µ(x, y)tcrk(x)
∑
z≥y
µ(y, z)tcrk(z)
=
∑
y∈L
tcrk(y)χ(Ly, t)
∑
x≤y
µ(x, y)trk(L)−rk(x)
=
∑
y∈L
tcrk(y)χ(Ly, t)trk(L)
∑
x≥opy
µop(y, x)t−rk(x)
=trk(L)
∑
y∈L
tcrk(y)χ(Ly, t)χ((Lop)y, t−1).
We can use Proposition 6.10 to compute M(P, t) for cases where χ(P, t) is well
known. Let Lnq be the modular lattice of all subspaces in Fnq , a vector space of dimen-
sion n over a field with q elements. The Möbius function and the characteristic polynomial
of Lnq are well known.
Proposition 6.11 ([34, Proposition 7.5.3]). In Lnq we have
µ(0̂, 1̂) = (−1)nq(
n
2)
and
χ(Lnq , t) =
n−1∏
i=0
(t− qi).
Using this we can get a nice formulation for M(Lnq , t). First we need to recall some
terminology from q-series. Let[
n
k
]
q
=
(qn − 1) · · · (q − 1)
(qk − 1) · · · (q − 1) · (qn−k − 1) · · · (q − 1)
be the q-binomial coefficient (aka Gaussian coefficient). Also, we denote by[
n
k1, k2, . . . , km
]
q
=
[
n
k1
]
q
[
n− k1
k2
]
q
· · ·
[
n− (k1 + · · · km−1)
km
]
q
the q-multinomial coefficient. We also use the q-Pochhammer symbol
(a; q)n =
n−1∏
i=0
(1− aqi).
224 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
We use [3] for a general reference for q-series. Using Proposition 6.11 we get the following.
Proposition 6.12. If Lnq is the modular lattice of subspaces of Fnq then
M(Lnq , t) =
∑
0≤i≤j≤k≤n
(−1)k−i
[
n
i, j − i, k − j, n− k
]
q
q(
j−i
2 )+(
k−j
2 )t3n−i−j−k.
Proof. Use that
[
n
k
]
q
counts the number of subspaces of dimension k in Fnq and apply
Theorem 5.2 to J in M(Lnq , t) together with Proposition 6.11.
Now we can reformulate Proposition 6.12 using Proposition 6.10 together with Propo-
sition 6.11 to get a nice identity in q-series.
Proposition 6.13. If Lnq is the modular lattice of subspaces of Fnq , then
M(Lnq , t) = tn
∑
0≤k≤n
tn−k
[
n
k
]
q
n−k−1∏
i=0
(t− qi)
k−1∏
j=0
(t− qj).
It turns out that −1 is a root of M(Lnq , t). We need a few results in order to prove this.
First we present a formula or q-identity which seems to be a kind of q-generalized binomial
theorem (the authors could not find it in the literature). It’s interesting that in the odd case
the sum trivially collapses but not for the even case.
Lemma 6.14. If n > 0 then
n∑
k=0
(−1)k
[
n
k
]
q
(−1 : q)n−k(−1; q)k = 0.
Proof. Let
S(n) =
n−1∑
k=0
(−1)k
[
n
k
]
q
(−1; q)n−k(−1; q)k
(−1; q)n
which is the left hand side up to the n − 1 term divided by the nth term. Using tech-
niques from [24] and Mathematica [15] we build a recursion for S(n). We compute
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 225
(1 + qn−1)S(n) =
n−1∑
k=0
(−1)k
(
qk
[
n− 1
k
]
q
+
[
n− 1
k − 1
]
q
)
(−1; q)n−k(−1; q)k
(−1; q)n−1
=
n−1∑
k=0
(−1)k
[
n− 1
k
]
q
(−1; q)n−k−1(−1; q)k
(−1; q)n−1
(qk + qn−1)
+
n−1∑
k=1
(−1)k
[
n− 1
k − 1
]
q
(−1; q)n−k(−1; q)k
(−1; q)n−1
=(−1)n−12qn−1 +
n−2∑
k=0
(−1)k
[
n− 1
k
]
q
(−1; q)n−k−1(−1; q)k
(−1; q)n−1
qk
+
n−2∑
k=0
(−1)k
[
n− 1
k
]
q
(−1; q)n−k−1(−1; q)k
(−1; q)n−1
qn−1
+
n−2∑
k=0
(−1)k+1
[
n− 1
k
]
q
(−1; q)n−(k+1)(−1; q)k+1
(−1; q)n−1
=(−1)n−12qn−1 +
n−2∑
k=0
(−1)k
[
n− 1
k
]
q
(−1; q)n−k−1(−1; q)k
(−1; q)n−1
qk
+ qn−1S(n− 1)−
n−2∑
k=0
(−1)k
[
n− 1
k
]
q
(−1; q)n−k−1(−1; q)k
(−1; q)n−1
(1 + qk)
=(−1)n−12qn−1 + qn−1S(n− 1)− S(n− 1).
Now we prove with induction that S(n) = (−1)n−1. First we see that S(1) = 1. Then
using the recursion above we have
(1 + qn−1)S(n) = (−1)n−12qn−1 − qn−1(−1)n−1 + (−1)n−1 = (−1)n−1(qn−1 + 1)
which finishes the proof.
Proposition 6.15. If Lnq is the modular lattice of subspaces of Fnq then M(Lnq ,−1) = 0.
Proof. Evaluate the expression in Proposition 6.13 and apply Lemma 6.14.
Now we can prove the main result of this section.
Theorem 6.16. If L is a modular geometric lattice (modular matroid) then M(L,−1) = 0.
Proof. Use the classical result that a modular geometric lattice is product of Boolean and
projective spaces (see 12.1 Theorem 4 in [32] or Proposition 6.9.1 in [23]). Then the result
follows from Propositions 6.15, 6.8, and 6.6.
Remark 6.17. The proof of Theorem 6.16 is done in cases. It would be interesting if there
was a case free proof just using the modular property.
226 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Remark 6.18. At first when looking at examples of M on the lattice of flats L(M) of
a matroid M it seems that the converse of Theorem 6.16 might be true. As for even the
simplest non-modular matroid U3,4 has M polynomial
M(L(U3,4), t) = (t− 1)(t8 − 3t7 − t6 + 12t5 − 2t4 − 12t3 + 3t2 + 5t− 1)
which does not have a factor of (t + 1). However, the converse is false, but the example
seems rather special. Using the SageMath computer algebra system [10] we compute
M(L(M∗(K3,3)), t) = (t10 − 9t9 + 22t8 + 12t7 − 81t6 + 21t5
+ 69t4 − 18t3 − 34t2 + 15t− 1)(t+ 1)(t− 1)
where M∗(K3,3) is the dual matroid of the graphic matroid corresponding to the complete
bipartite graph K3,3. Since M∗(K3,3) is a connected non-modular matroid (it does not
have a modular direct summand) this example gives a connected non-modular matroid that
has −1 as a root of M. This example and Theorem 6.16 motivate a few questions.
Question 6.19. Is there a rank 3 non-modular connected matroid M such that M(M,−1) =
0?
Question 6.20. Is there a classification of all matroids whose M polynomial has -1 as a
root?
Question 6.21. Is there a nice enumerative combinatorial interpretation for M(M,−1)
where M is a matroid (i.e. what does it count)?
6.1 No Deletion-Contraction
We now show that J and M are not some evaluation of the Tutte polynomial for matroids.
We first recall the following definition.
Definition 6.22. We say that a function f from matroids to a ring R is a generalized Tutte-
Grothendieck invariant (following [4] Sec 1.8.6) if there exists a, b ∈ R such that for every
matroid M and element of the ground set e ∈ M
f(M) =

f(M\e)f(L) if e is a loop
f(M/e)f(C) if e is a coloop
af(M\e) + bf(M/e) otherwise.
where L is the matroid consisting of exactly one loop and C is the matroid consisting of
exactly one coloop.
Let Ur,n be the uniform matroid of rank r on n elements and recall that Ur,r ∼= Br are
Boolean or free matroids. Then, a direct computation gives J (B1, t) = t+ 1 and
J (U2,n, t) = (n− 1)t2 + nt+ n− 1.
Hence J(U2,3, t) = 2t2 + 3t + 2. Then any deletion is U2,3\e ∼= B2 and any contraction
is U2,3/e ∼= U1,2. Putting this together with Definition 6.22 and assuming that J is a
Tutte-Grothendieck invariant
2t2 + 3t+ 2 = a(t2 + 2t+ 1) + b(t+ 1).
J. Johnson et al.: A non-associative incidence near-ring with a generalized Möbius function 227
However, this is a contradiction since t+ 1 is not a factor of the left hand side.
The same result for M needs two more steps. Looking at the same matroid and using
Proposition 6.9 we get
M(U2,3, t) = (t2 − 3t+ 1)(t+ 1)2(t− 1)2 = a(t+ 1)2(t− 1)4 + b(t+ 1)(t− 1)2
which reduces to
b = (t+ 1)(t2 − 3t+ 1)− a(t+ 1)(t− 1)2.
Then we look at U2,4 and again assume M is a Tutte-Grothendieck invariant
M(U2,4, t) = (t2−4t+1)(t+1)2(t−1)2 = a(t2−3t+1)(t+1)2(t−1)2+b(t+1)(t−1)2.
Inserting the above value for b and reducing we get
t2 − 4t+ 1 = a(t2 − 3t+ 1) + (t2 − 3t+ 1)− a(t− 1)2
which gives a = 1 and makes b = −t(t+ 1). But then
M(U3,4, t) = (t− 1)(t8 − 3t7 − t6 + 12t5 − 2t4 − 12t3 + 3t2 + 5t− 1)
which does not have a factor of t+ 1. This is a contradiction since the right hand side
M(U3,4\e, t)− t(t+ 1)M(U3,4/e, t) = M(U3,3, t)− t(t+ 1)M(U2,3, t)
does have a t+ 1 factor.
6.2 Valuations
Here we study the invariant M over matroid subdivisions. One could focus on a wider
range combinatorial objects like posets but we are motived by applications to matroid the-
ory. First we recall the basis matroid polytope (using [6] as our general reference for this
material). A matroid M can be defined via its set of bases B(M) which are all the inde-
pendent sets of M whose size is the rank of M . Then, the matroid polytope of M is
P (M) = Conv{eB |B ∈ B(M)}
where eB = ei1 + · · ·+ eir with B = {i1, . . . , ir}. Now we need a few key definitions to
state our main result.
Definition 6.23. A matroid polyhedral subdivision of a matroid polytope P (M) is a col-
lection of polyhedra {Pi} such that
⋃
Pi = P (M), each Pi is a matroid polytope whose
vertices are vertices of P (M), and for i ̸= j if Pi
⋂
Pj ̸= ∅, then Pi
⋂
Pj is a proper face
of both Pi and Pj .
Now we want to know how invariants decompose across subdivisions which gives rise
to valuations. We will use what is called a weak valuation in [6] but we follow [5] and just
say valuation. This makes sense since by Theorem 4.2 in [6] for matroids weak valuations
are actually strong valuations.
228 Ars Math. Contemp. 24 (2024) #P2.03 / 207–230
Definition 6.24. Let P be the collection of matroid polytopes and R a commutative ring.
A function f : P → R is a (weak) valuation if for any matroid polytope P (M) and any
matroid polyhedral subdivision with maximal pieces {P (M1), . . . , P (Mk)} we have that
f(∅) = 0 and
f(P (M)) =
∑
{j1,...,ji}⊆[k]
(−1)if(P (Mj1) ∩ · · · ∩ P (Mji)).
Finally we can state the result for the invariant J in terms of valuations.
Proposition 6.25. The polynomial J is a valuation on matroids.
Proof. Using the decomposition of the J-function given in Theorem 5.2 we know that
J (M, t) = (−1)rk(M)
∑
X∈L(M)
µ(∅, X)µ(X, 1̂)trk(X)
where 1̂ is the maximal flat of M . Hence as a function from the collection of matroids to
Z[t] we can represent the function J as
J = (±1)
∑
f1 ⋆ f2
where f1 ⋆ f2 = m ◦ (f1 ⊗ f2) ◦ ∆S,T from the notation in Theorem C in [6] and f1 =
χM (0) and f2 = χM (0)trk(M). Since f1 and f2 are both Tutte-Grothendieck invariants for
matroids and are evaluations of the Tutte polynomial we can conclude that f1 and f2 are
both valuations from Proposition 7.5 in [6]. Finally putting it all together Theorem C in [6]
finished the result.
We conclude with a natural question. The polynomial M(L, t) is slightly more com-
plicated but has promising properties that seems to imply it should be a valuation.
Question 6.26. Is the polynomial M a matroid valuation? It seems that Proposition 6.10
with Proposition 7.5 and Theorem C in [6] is essentially the proof. However that would
use that the characteristic polynomial on the flipped lattice of flats L(M)op is a matroid
valuation.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.04 / 231–271
https://doi.org/10.26493/1855-3974.2968.d23
(Also available at http://amc-journal.eu)
Valuations and orderings
on the real Weyl algebra
Lara Vukšić *
Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia and
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Received 19 September 2022, accepted 12 February 2023, published online 20 September 2023
Abstract
The first Weyl algebra A1(k) over a field k is the k-algebra with two generators x, y
subject to [y, x] = 1 and was first introduced during the development of quantum mechan-
ics. In this article, we classify all valuations on the real Weyl algebra A1(R) whose residue
field is R. We then use a noncommutative version of the Baer-Krull theorem to classify all
orderings on A1(R). As a byproduct of our studies, we settle two open problems in real
algebraic geometry. First, we show that not all orderings on A1(R) extend to an ordering
on a larger ring R[y; δ], where R is the ring of Puiseux series, introduced by Marshall and
Zhang in 2000, and characterize the orderings that do have such an extension. Second,
we show that for valuations on noncommutative division rings, Kaplansky’s theorem that
extensions by limits of pseudo-Cauchy sequences are immediate fails in general.
Keywords: Weyl algebra, noncommutative valuations, skew polynomial rings, orderings, extensions
of valuations, extensions of orderings.
Math. Subj. Class. (2020): 16W60, 06F25, 13J30, 14A22, 16S3
1 Introduction
Valuation theory was first developed for commutative fields in the context of number theory
and was first defined by József Kürschák [12] in 1913. For modern treatments, we refer to
the books of Engler and Prestel [5] or Kuhlmann [10]. Oscar Schilling wrote the first major
work on valuations on (noncommutative) division rings in 1945 [21].
A valuation on a division ring D is a map v : D → Γ ∪ {∞}, where Γ is an ordered
group written additively and ∞ ̸∈ Γ,∞ > γ for each γ ∈ Γ, with the following properties:
*I would like to thank my advisor Igor Klep for his guidance and many helpful comments and suggestions.
E-mail address: lara.vuksic@fmf.uni-lj.si (Lara Vukšić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
232 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
1. ∀x ∈ D : v(x) = ∞ ⇔ x = 0,
2. ∀x, y ∈ D : v(xy) = v(x) + v(y),
3. ∀x, y ∈ D : v(x+ y) ≥ min{v(x), v(y)}.
It follows that v is a homomorphism from D∗ to Γ. The set Ov := {x ∈ D | v(x) ≥ 0} is
called the valuation ring associated to v, and Mv := {x ∈ M | v(x) > 0} is its maximal
ideal. The division ring D := Ov/Mv is called the associated residue division ring. Since
v is a group homomorphism, the subgroup O∗v is normal in D∗. Several alternative ap-
proaches to noncommutative valuations, where v does not define a group homomorphism,
were introduced and studied recently by Nicolai Ivanovich Dubrovin in [7] and [4] (see
also [17] for a more thorough treatment), and by Jean-Pierre Tignol and Adrien Wadsworth
in [23].
Suppose F is a field. Then all valuations on the field or rational functions F (x) with
residue field F are well-known, namely, the p-adic valuations for irreducible polynomials
p(x) ∈ F [x], and the vdeg valuation, defined by
vdeg(
p
q
) := deg(q)− deg(p).
The description of all valuations on the field of rational functions in several variables with
residue field equal to the base field is much more involved. There are many descriptions
of constructions of such valuations in the literature. Among famous examples of such
descriptions are the one given by Saunders MacLane in [13] and the one given by Franz-
Viktor Kuhlmann in [11].
As valuations on Ore extensions uniquely extend to their quotient division ring, the
description of all valuations on Ore division rings is equivalent to the description of all
valuations on corresponding quotient division rings. The description of all valuations on
noncommutative Ore extensions R[x;σ, δ] where R is a domain, σ : R → R is a ring
homomorphism and δ : R → R a σ-derivation is even more complex than in the commu-
tative case. Additional difficulties arise from the fact that [f, g] = 0 does not hold for all
f, g ∈ R[x;σ, δ]. Granja, Martı́nez, and Rodrı́guez have shown in [6] that the set of all
real valuations extending to the skew polynomial ring has the structure of a parameterized
complete non-metric tree. Further recent progress on valuations on Ore extensions is given
by Onay in [18] and Rohwer in his PhD thesis [20].
1.1 Results
Our main goal is to classify all orderings and real valuations on the real Weyl algebra A1(R)
or, equivalently, its quotient division ring D1(R). The Weyl algebra is the noncommutative
algebra generated by two elements x, y satisfying [y, x] = 1. Hence its elements are all of
the form ∑
i,j
αi,jx
iyj , αi,j ∈ R.
Because of this, our approach to constructing valuations on A1(R) is inspired by classical
constructions of valuations on commutative rational functions in two unknowns mentioned
above. However, the relation [y, x] = 1 gives rise to additional constraints and far fewer
valuations than in the commutative case.
L. Vukšić: Valuations and orderings on the real Weyl algebra 233
As we will show, the valuations on A1(R) we are interested in all satisfy v[a, b] > v(ab)
for all nonzero a, b. We call such valuations strongly abelian. They have an abelian value
group and commutative residue field. In Section 2 we give some properties of strongly
abelian valuations. We show that if a valuation v on a division ring D satisfies D = Z(D)
and the value group is of rational rank one, then v is strongly abelian. Under additional
constraints on the residue field and the value group we extend this statement to valuations
of higher rational rank.
In Section 3 we give a characterization of all valuations v on the real Weyl algebra
A1(R) with residue field R in the spirit of MacLane [13]. The construction is inspired
by the outline given by Shtipelman in [22] for valuations on the complex Weyl algebra
A1(C). We also explicitly describe the associated value groups and show that they are all
isomorphic to subgroups of Q or Q× Z.
In their attempt to describe all orderings on A1(R) in [16], Murray Marshall and Yufei
Zhang introduced the Ore extensions R[y; δ] and R̃[y; δ], with
R := {
∑
k≥m
akx
− kn | ak ∈ R,m ∈ Z, n ∈ N},
R̃ := {
∑
q∈A
aqx
−q | A ⊂ Q is well-ordered}
and δ(p(x)) = p′(x). As is often done in real algebraic geometry, all orderings are de-
scribed by classifying all real valuations via the Baer-Krull theorem. Marshall and Zhang
described almost all valuations v on R[y; δ] with residue field R; in one case, they did not
prove that v is a valuation. In Section 4, we complete their characterization. Marshall
and Zhang also conjectured that all valuations on A1(R) with residue field R extend to a
valuation on R[y; δ] with the same residue field. We refute their conjecture in Section 4.
Further, we combine our classification of valuations on A1(R) with Marshall and Zhang’s
description of valuations on R[y; δ] to characterize the valuations on A1(R) with residue
field R that extend to a valuation R[y, δ] with the same residue field. All such extensions
are again strongly abelian.
In Section 5, we show that all valuations on R[y; δ] with residue field R uniquely extend
to a strongly abelian valuation on R̃[y; δ] with the same residue field. We also show that
the value group of such an extension is not of rational rank one.
As a byproduct of our investigations, we show that Kaplansky’s theorem that all ex-
tensions by limits of pseudo-Cauchy sequences are immediate (in particular, they do not
change the rational rank of the value group) fails for noncommutative division rings.
As Marshall and Zhang observe in [15], all strongly abelian valuations v on a division
ring D with a formally real residue field are compatible with an order on D. In Section 6,
we describe all v-compatible orders on A1(R) for every valuation v on A1(R) constructed
in Section 3 using a noncommutative version of the Baer-Krull theorem as given in [2]
(see also [1, 3, 24] and [9] for modern treatments and extensions). We also characterize
the v-compatible orders on A1(R) that extend to an order on R[y; δ] compatible with v’s
extension to R[y; δ].
2 Strongly abelian valuations
We present some properties of valuations on noncommutative division rings which we will
use later to describe order-compatible valuations on the real Weyl algebra A1(R) and some
of its ring extensions. First, we define a property of valuations on division rings.
234 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
Definition 2.1. Suppose v is a valuation on a division ring D. We say v is strongly abelian
if v[a, b] > v(ab) holds for all nonzero a, b ∈ D.
Any valuation on a field is strongly abelian. In this section, we describe a sufficient
condition for a valuation v to be strongly abelian. This property will be important for us
for two reasons. Firstly, it is obvious that if a valuation v on a division ring D is strongly
abelian, then the associated value group and residue division ring are commutative. Sec-
ondly, we are particularly interested in order-compatible valuations on A1(R); minimal
such have residue field R, as it was shown in [16]. It follows from Theorem 2.5 of [15]
that a strongly abelian valuation v on a division ring D with a formally real residue field is
compatible with an order on D by the noncommutative version of the Baer-Krull theorem
as given in [24].
Proposition 2.2. Let v be a valuation on a division ring D such that D = Z(D). Let
a, b ∈ D∗ be such that v(a) and v(b) are rationally dependent. Then v[a, b] > v(ab).
Proof. Since v(a) and v(b) are rationally dependent, v(ab) = v(ba) ≤ v[a, b]. Suppose
v[a, b] = v(ab). In particular, a and b do not commute. Let β := aba−1b−1 ∈ D. We have
β = aba−1b−1 = ([a, b] + ba)a−1b−1 = [a, b]a−1b−1 + 1 ̸= 1.
Let v(b) = − ℓkv(a) for ℓ, k ∈ Z, ℓ and k coprime. It follows that v[a, b] =
k−ℓ
k v(a).
Define γ := (ba)kaℓ−k ∈ D. Let β′, γ′ ∈ Z(D) be such that v(β′) = v(γ′) = 0, β′ = β
and γ′ = γ. Then on one hand,
v(a(ba)k − γ′ak−ℓ+1) = v(a((ba)kaℓ−k − γ′)ak−ℓ) > (k − ℓ+ 1)v(a),
and on the other,
v(a(ba)k − γ′ak−ℓ+1) = v((ab)ka− γ′ak−ℓ+1) = v(((ab)kaℓ−k − γ′)ak−ℓ+1)
= (k − ℓ+ 1)v(a)
since
(ab)kaℓ−k − γ′ = (aba−1b−1ba)kaℓ−k − γ = βk(ba)kaℓ−k − γ ̸= 0.
In the last equation, we used that (aba−1b−1ba)kaℓ−k = (β′ba)kaℓ−k = β′k(ba)kaℓ−k =
βk(ba)kaℓ−k since D = Z(D) as presumed. This holds if βk ̸= 1. If βk = 1, we repeat
our calculations with roles of a and b interchanged. The new β will now be the inverse value
of the former and since gcd(ℓ, k) = 1, β−ℓ ̸= 1. In either case, we get a contradiction from
which we deduce v[a, b] > v(ab).
Remark 2.3. The condition D = Z(D) is fulfilled by every valuation on an algebra over
a field that is isomorphic to the residue field. In particular, this holds for minimal order-
compatible valuations on R-algebras.
Corollary 2.4. Let v be a valuation on a division ring D such that D = Z(D). If the value
group has rational rank one, then v is strongly abelian.
L. Vukšić: Valuations and orderings on the real Weyl algebra 235
Lemma 2.5. Let v be a valuation on a division ring D such that the value group is abelian
and D = Z(D). Then for all x, y ∈ D \ {0}:
(1) If v[x, y] > v(xy), then v[xm, y] > v(xmy) for all m ∈ Z.
(2) Suppose v[x, y] = v(xy). Then v[x−1, y] = v(x−1y) and for each m ∈ N,
v[xm, y] > v(xmy) holds if and only if α := y−1x−1yx satisfies 1 + α + · · · +
αm−1 = 0 in D.
Proof. To prove (1), first observe
[x−1, y] = x−1y − yx−1 = x−1(yx− xy)x−1,
so if v[x, y] > v(x, y), then
v[x−1, y] = v[x, y]− 2v(x) > v(xy)− 2v(x) = v(x−1y).
Suppose m ∈ N. Then
[xm, y] =
m∑
ℓ=1
xm−ℓ[x, y]xℓ−1
and since the value group is commutative, v(xm−ℓ[x, y]xℓ−1) = (m− 1)v(x) + v[x, y] >
v(xmy) for each 1 ≤ ℓ ≤ m. Item (1) is thus proved.
To prove (2), suppose v[x, y] = v(xy). Then v[x−1, y] = v(x−1y) is proved as for the
first case. Since the value group is abelian, v[xm, y] ≥ v(xmy) holds, so we can observe
y−1x−m[xm, y] = y−1x−m
m∑
ℓ=1
xm−ℓ[x, y]xℓ−1 =
m∑
ℓ=1
y−1x−ℓ[x, y]xℓ−1.
For each 1 ≤ ℓ ≤ m,
y−1x−ℓ[x, y]xℓ−1 = (αx−1)ℓ−1y−1x−1[x, y]xℓ−1
= αℓ−1x−ℓ+1y−1x−1[x, y]xℓ−1 = y−1x−1[x, y]αℓ−1.
We can change the order of α and x−1 by Proposition 2.2 since v(α) = 0. The last equation
follows from v(y−1x−1[x, y]) = 0 and Proposition 2.2. So now we have
y−1x−m[xm, y] = y−1x−1[x, y]
m−1∑
ℓ=0
αℓ,
which proves the equivalence in (2).
Proposition 2.6. Let v be a valuation on a division ring D such that the value group is
abelian and D = Z(D). Suppose the residue field is formally real and suppose v[x, y] =
v(xy) for some x, y ∈ D. Then v[xm, y] = v(xmy) for all odd m > 2. If v[xm, y] >
v(xmy) for some even m, then v[x2, y] > v(x2y) and there is no a ∈ D such that v(a2) =
v(x).
236 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
Proof. Suppose v[x, y] = v(xy). If m is odd,
∑m−1
ℓ=0 α
ℓ = 0 does not have a solution in
the residue field. By Lemma 2.5, it follows that v[xm, y] = v(xmy). Now consider the case
for even m. If v[xm, y] > v(xmy), then α = −1 by Lemma 2.5 and v[x2, y] > v(x2y).
Suppose a ∈ D satisfies v(a2) = v(x). We will first show that v[a2, y] = v(a2y). Assume
that v[a2, y] > v(a2y). Then on one hand,
a−2x = a−2xy−1y = y−1a−2xy,
since v(a−2x) = 0. On the other hand,
a−2x = y−1ya−2x = y−1a−2yx,
where the last equation follows from v[a2, y] > v(a2y), or, by Lemma 2.5 equivalently,
v[a−2, y] > v(a−2y). From x−1a−2(xy − yx) = 0 we conclude v[x, y] > v(xy), which
is a contradiction. So v[a2, y] = v(a2y) and v[a, y] = v(ay) follows from Lemma 2.5.
Now we show v[a4, y] > v(a4y). On one hand, we can write
a−4x2 = y−1a−4x2y = y−1a−4yx2
since v[x2, y] > v(x2y). On the other hand,
a−4x2 = y−1ya−4x2,
so we conclude v[a4, y] > v(a4y). But by Lemma 2.5, v[a4, x] > v(a4x) gives us
xax−1a−1 = −1. But then, again by Lemma 2.5, v[a2, x] > v(a2x). The proposition
is thus proved.
Proposition 2.7. Let v be a valuation on a division ring D such that such that D = Z(D),
the value group is abelian and 2-divisible and the residue field is formally real. Suppose
the value group of v is of rational rank 2 and suppose there are x, y ∈ D∗ such that v(x)
and v(y) are rationally independent with v[x, y] > v(xy). Then v is strongly abelian.
Proof. Suppose a, b ∈ D. Since the value group is abelian, v[a, b] ≥ v(ab). Suppose
v[a, b] = v(ab). Then v(ak1) = v(x−m1y−n1) and v(bk2) = v(x−m2y−n2) for some
ki,mi, ni ∈ Z, i = 1, 2. We conclude from Lemma 2.5 that v[ak1 , bk2 ] = v(ak1bk2).
This is immediate if k1 and k2 are both odd. If k1 or k2 is even, v[ak1 , bk2 ] = v(ak1bk2)
follows from the 2-divisibility of the value group and Proposition 2.6. Let c := xm1yn1
and d := xm2yn2 . Then on the one hand,
ak1cbk2d = cak1dbk2 = dcak1bk2
since v(ak1) and v(c) are rationally dependent, v(bk2) and v(d) are rationally dependent
and v(bk2d) = v(ak1c) = 0. On the other hand,
ak1cbk2d = bk2dak1c = dbk2cak1 = cdbk2ak1 = dcbk2ak1 .
Here, the last equality follows from v[x, y] > v(xy) and Lemma 2.5. Thus we have
v(dc(ak1bk2 − bk2ak1)) > 0, so we get v[ak1 , bk2 ] > v(ak1bk2) which contradicts our
assumption v[a, b] = v(a, b). We conclude v[a, b] > v(ab).
The proof of the following proposition is the same as the proof of Proposition 2.7.
L. Vukšić: Valuations and orderings on the real Weyl algebra 237
Proposition 2.8. Let v be a valuation on a division ring D such that D = Z(D), the
value group is abelian and the residue field is formally real. Suppose the value group
of v is of rational rank 2 and suppose there are x, y ∈ D∗ such that for every z ∈ D,
v(zk) = v(x−my−n) holds for some k,m, n ∈ Z where k is odd. Then v is strongly
abelian.
We will later use this result to show that all valuations v on A1(R) with residue field R
are strongly abelian. Propositions 2.7 and 2.8 can be easily generalized to higher rational
ranks of the value group. The proofs are analogous.
Corollary 2.9. Let v be a valuation on a division ring D such that D = Z(D). Suppose the
value group is abelian and 2-divisible of rational rank n and that there are x1, . . . , xn ∈ D
such that v(x1), . . . , v(xn) are rationally independent with v[xi, xj ] > v(xixj) for all i, j.
Then v is strongly abelian.
Corollary 2.10. Let v be a valuation on division ring D such that D = Z(D). Suppose
the value group is abelian and of rational rank n and that there are x1, . . . , xn ∈ D such
that for every z ∈ D, v(zk) = v(xm11 · · ·xmnn ) for some k,m1, . . . ,mn ∈ Z with k odd.
Then v is strongly abelian.
3 Valuations on A1(R)
We now describe the construction of all valuations on A1(R) with residue field R that was
sketched in [22] over the ground field of C. Since every f ∈ A1(R) can be written as∑
m,n≥0 αm,nx
myn, the construction will be similar to the construction of all valuations
on the field of rational functions R(x, y) with residue field R (examples of constructions of
such valuations can be found in [11] or [13]), but with some additional constraints arising
from the fact that the generators x, y ∈ A1(R) satisfy [y, x] = 1. We first note that
it follows from Theorem 5.3 of [16] that the value group of any valuation on A1(R) is
commutative. Also, since every valuation v on A1(R) can be uniquely extended to its
quotient division ring D1(R), our construction will take place in the quotient ring as we
will use inverses.
To construct a valuation v trivial on R with residue field R, we compare v(x) and v(y).
It is easy to show, as it was done in [16], that v(xy) = v(yx) < 0, so v(x) or v(y) will be
less than zero. Without loss of generality, we can set v(x) = −1 ∈ Q and compare it to
v(y). If v(y) ̸∈ Q, then we get
v(
∑
m,n≥0
αm,nx
myn) = min
m,n
{mv(x) + nv(y)}
for all elements of A1(R). Otherwise, v(y) = m1n1 ∈ Q. It follows that x
m1yn1 = β1 ∈ R,
so v(xm1yn1 − β1) > 0. Set
ω1 := x
m1yn1 − β1
and as before compare v(ω1) to v(x) in terms of rational dependence. If v(ω1) = m2n2 ∈ Q,
then xm2ωn21 = β2 for some β2 ∈ R. Hence
ω2 := x
m2ωn21 − β2
238 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
also has value greater than zero. We continue this procedure. If we additionally define
ω−1 = x and ω0 = y, we thus get a sequence
(ωi)i≥−1, ωi ∈ A1(R)
which ends with ωn for some n ∈ N if v(ωn) ̸∈ Q or is infinite otherwise.
By the end of this section, we will prove a necessary and sufficient condition for the
possibility to extend v from (ωi)i≥−1 to a valuation on A1(R) with residue field R. Every
such extension from (ωi)i≥−1 to A1(R) will be uniquely determined. We will also show
that every valuation on A1(R) with residue field R is strongly abelian.
3.1 Properties of the sequence (ωi)i≥−1 associated to a valuation on A1(R)
Thorough this subsection let v be a valuation on A1(R).
Lemma 3.1. Suppose (ωi)i≥−1 ⊆ A1(R) is a sequence as described above, with ω−1 = x,
ω0 = y, ωi = xmiωnii−1 − βi for all i ≥ 0. Then [ωi, x] equals
xmi
ni∑
ℓi=1
ωni−ℓii−1
(
xmi−1
ni−1∑
ℓi−1=1
ω
ni−1−ℓi−1
i−2(
· · ·
(
xm2
n2∑
ℓ2=1
ωn2−ℓ21 n1x
m1yn1−1ωℓ21
)
ωℓ32
)
· · ·
)
ωℓii−1
for each i ≥ 1.
Proof. We prove the lemma by induction on i. If i = 1,
[ω1, x] = [x
m1yn1 − β1, x] = xm1
n1∑
ℓ1=1
yn1−ℓ1 [y, x]yℓ1−1 = n1x
m1yn1−1.
Now suppose that the equality holds for [ωi, x]. Then we have
[ωi+1, x] = [x
mi+1ω
ni+1
i − βi+1, x] = x
mi+1 [ω
ni+1
i , x]
= xmi+1
ni+1∑
ℓi+1=1
ω
ni+1−ℓi+1
i [ωi, x]ω
ℓi+1−1
i
We can then proceed by the induction hypothesis.
Before proving the next lemma, we define an equivalence relation between nonzero
elements of A1(R) that have the same v-value, but their difference does not. For any
a, b ∈ A1(R) \ {0}, we write a ∼ b if v(a) = v(b) < v(a− b). This is also a congurence
relation, as ac ∼ bc and ca ∼ cb holds for all a, b, c ∈ A1(R) \ {0} with a ∼ b.
Lemma 3.2. Suppose v is a valuation on A1(R) with residue field R and suppose (ωi)i≥−1
is a sequence such that ω−1 = x, ω0 = y, ωi = xmiωnii−1 − βi, v(x) = −1 and v(ωi) =
mi+1
ni+1
for all i ≥ 0 up to either some n ≥ 0 in which case v(ωn) ̸∈ Q, or up to infinity.
Then v[ωj , ωi] > v(ωiωj) for all i, j ≤ k if and only if v(Πkℓ=−1ωℓ) < 0, where k ≤ n in
case v(ωn) ̸∈ Q for some n ≥ 0.
L. Vukšić: Valuations and orderings on the real Weyl algebra 239
If any and hence both sides of the equivalence hold, then
v[ωj , ωi] = −v(xyω1 · · ·ωi−1ωi+1 · · ·ωj−1)
for all i < j ≤ k.
Proof. Suppose v is a valuation on A1(R) and (ωi)i≥−1 is a sequence as described in the
lemma. So v(ωi) ∈ Q either for all i ≥ 0 or for all 0 ≤ i < n for some n ≥ 0 and
v(ωn) ̸∈ Q. It follows from Proposition 2.2 that v[ωi, ωj ] > v(ωiωj) for all i, j < n since
v(ωi) and v(ωj) are rationally dependent. We shall use this fact to evaluate v[ωi, ωj ] for all
i, j ≤ k ≤ n. It follows from Lemma 3.1 that [ωk, x] is a sum of products P , all equal to
yn1−1xm1ωn2−11 x
m2 · · ·ωnk−1k−1 xmk up to the order of factors. Since v[ωi, ωj ] > v(ωiωj)
for all i, j ≤ k − 1,
P ∼ yn1−1xm1ωn2−11 xm2 · · ·ω
nk−1
k−1 x
mk
holds for every product P of the sum. Since
v(yn1−1xm1ωn2−11 x
m2 · · ·ωnk−1k−1 x
mk) + v(yω1 · · ·ωk−1) =
k∑
i=1
v(xmiωnii−1) = 0,
we can conclude v[ωk, x] = v(yn1−1xm1ωn2−11 x
m2 · · ·ωnk−1k−1 xmk) = −v(yω1 · · ·ωk−1).
It follows that v[x, ωk] > v(xωk) if and only if v(xyω1 · · ·ωk) < 0.
We will now prove that v[ωi+k, ωi] = −v(xyω1 · · ·ωi−1ωi+1 · · ·ωi+k−1) by induction
on k, 1 ≤ k ≤ n − i. It will then follow that v[ωi, ωj ] > v(ωiωj) for all i, j ≤ n if and
only if v(Πnℓ=−1ωℓ) < 0. If k = 1, then
[ωi+1, ωi] = [x
mi+1ω
ni+1
i − βi+1, ωi] = [x
mi+1 , ωi]ω
ni+1
i
= (
mi+1∑
ℓ=1
xmi+1−ℓ[x, ωi]x
ℓ−1)ω
ni+1
i ,
and since [x, ωi] is a sum of products all equal to yn1−1xm1ωn2−11 x
m2 · · ·ωni−1i−1 xmi up to
the order of factors, we can, using v[ωi, ωj ] > v(ωiωj) for j < i, deduce that v[ωi+1, ωi] =
−v(xyω1 · · ·ωi−1) just like we did when evaluating v[ωi, x]. For k > 1, we have
[ωi+k, ωi] = [x
mi+kω
ni+k
i+k−1 − βi+k, ωi] = [x
mi+k , ωi]ω
ni+k
i+k−1 + x
mi+k [ω
ni+k
i+k−1, ωi]
= (
mi+k∑
ℓ=1
xmi+k−ℓ[x, ωi]x
ℓ−1)ω
ni+k
i+k−1+
xmi+k(
ni+k∑
ℓ=1
ω
ni+k−ℓ
i+k−1 [ωi+k−1, ωi]ω
ℓ
i+k−1)
∼ mi+kxmi+k−1ω
ni+k
i+k−1[x, ωi] + ni+kx
mi+kω
ni+k−1
i+k−1 [ωi+k−1, ωi]
and using both Lemma 3.1 and induction on k, we see that the first sum has v-value equal
to −v(xyω1 · · ·ωi−1) and the second has v-value equal to −v(xyω1 · · ·ωi+k−1). Since the
latter is smaller, it is equal to v[ωi+k, ωi]. This proves the lemma.
240 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
It follows that if v can be extended from (ωi)i≥−1 to a valuation on A1(R),∑k
i≥−1 v(ωi) must be strictly less than 0 for all k ≤ n in case v(ωn) ̸∈ Q for some n,
and for all k ≥ 0 if v(ωi) ∈ Q for all i ∈ N. We will now describe a necessary condition
for the residue field to be R and then proceed to show that if both conditions are fulfilled,
v can be extended from (ωi)i≥−1 to a valuation on A1(R) with residue field R.
To ensure that the residue field is R, it is obviously necessary that ωkii−1ω
kj
j−1 ∈ R holds
for all ki, kj ∈ Z with ki mini + kj
mj
nj
= 0. For given i, j ≥ 0, all solutions (ki, kj) ∈ Z2 to
the diophantine equation
kimjni + kjminj = 0 (3.1)
are integer multiples of the pair (Ki,j ,−Kj,i) with
Ki,j =
mjni
di,j
,
Kj,i =
minj
di,j
,
where di,j = gcd{mjni,minj}.
So for all ki, kj ∈ Z with ki mini + kj
mj
nj
= 0, we can write
ωkii−1ω
kj
j−1 = ω
nKi,j
i−1 ω
−nKj,i
j−1 = (ω
Ki,j
i−1 ω
−Kj,i
j−1 )
n
for some n ∈ Z, where we used Proposition 2.2 in the second equality. So for every
ki, kj ∈ Z satisfying 3.1, ωkii−1ω
kj
j−1 is uniquely determined by ω
Ki,j
i−1 ω
−Kj,i
j−1 . For each
i, j ≥ 0, we define αi,j = ω
Ki,j
i−1 ω
−Kj,i
j−1 . We immediately see that αj,i = α
−1
i,j and hence
αi,i = 1 for all i, j ≥ 0. As
α
di,j
i,j = ω
Ki,jdi,j
i−1 ω
−Kj,idi,j
j−1 = ω
mjni
i−1 ω
−minj
j−1 = β
mj
i β
−mi
j
for all i, j ≥ 0, αi,j is one of the possible di,j-th roots for β
mj
i β
−mi
j . If v is a valuation on
D1(R) with residue field R, αi,j must be real for all i, j ≥ 0. For every i, j ≥ 0 with even
di,j , this means that β
mj
i β
−mi
j > 0 must hold. In the next lemma, we present a necessary
condition on the sequence (βi)i≥1 so that αi,j ∈ R can be chosen for all i, j. We also prove
that if ni is odd, αi,j is uniquely determined for all j ≥ 0.
Lemma 3.3. Let v be a valuation on D1(R) as in Lemma 3.2. Then the following holds:
(1) If ni is odd, there is a unique possible choice for αi,j ∈ R for all j ≥ 0.
(2) Only if sgn(βi) is constant on the set of all i ≥ 0 for which ni is even can we choose
αi,j ∈ R for all i, j ≥ 0.
Proof. Suppose ni is odd. Then for any j ≥ 0, let d̃i,j be the highest odd number dividing
di,j . Since ni is odd, ℓ1 :=
d̃i,jmj
di,j
∈ Z. If nj is odd as well, ℓ2 := d̃i,jmidi,j ∈ Z holds too.
If nj is even, mj is odd, so di,j = ˜di,j as di,j divides mjni. In both cases, ℓ2 ∈ Z holds.
L. Vukšić: Valuations and orderings on the real Weyl algebra 241
Then for ℓ := ℓ1mi = ℓ2mj =
d̃i,jmimj
di,j
we can evaluate
α
d̃i,j
i,j = ω
Ki,j d̃i,j
i−1 ω
−Kj,id̃i,j
j−1 = x
ℓx−ℓωℓ1nii−1 ω
−ℓ2nj
j−1 = (x
miωnii−1)
ℓ1(xmjω
nj
j−1)
−ℓ2
= βℓ1i β
−ℓ2
j ,
and since d̃i,j is odd, αi,j ∈ R is uniquely determined. The first point of the lemma is thus
proven.
To prove the second point of the lemma, suppose i, j ≥ 0 are such that ni and nj are
both even. As a consequence, both mi and mj are odd while di,j is even. So, provided
αi,j ∈ R, we compute
1 = sgn(α
di,j
i,j ) = sgn(β
mj
i β
−mi
j ) = sgn(βiβj),
which proves the second part of the lemma.
For even di,j we have seemingly two choices for αi,j ∈ R – a positive and a nega-
tive one. We will show that in most cases, we cannot choose sgn(αi,j) for all i, j ≥ 0
independently of each other.
Before that, we observe that for any i, j ≥ 0, at most one of Ki,j and Kj,i is even. In
fact, if at most one of ni and nj is odd, Ki,j is odd if and only if ni is divisible by the
greatest power of two that divides nj . For each i ≥ 1, let 2hi be the biggest power of two
that divides ni. Define also m0 = 1 and n0 = −1.
Proposition 3.4. Let v be a valuation on D1(R) associated to a sequence (ωi)i≥−1 with
v(ω−1) = v(x) = −1, v(ωi−1) = mini with gcd(mi, ni) = 1 and x
miωnii−1 = βi ∈ R
for each i ≥ 1. Suppose sgn(βi) is constant on the set of all i ≥ 0 for which ni is
even. Suppose αi,j ∈ R is determined for all i, j ≥ 0. Then Πri=0ω
ki
i−1 ∈ R is uniquely
determined for each set of integers k0, k1, . . . , kr ∈ Z with
∑r
i=0 ki
mi
ni
= 0 if and only if
for each a, b, c ≥ 0, αa,bαa,cαb,c > 0 whenever ha = hb ≤ hc holds.
Proof. To prove the necessity of the condition, suppose a, b, c ≥ 0 are such that ha =
hb ≤ hc holds. Suppose Ka,b and Kb,c are both odd. Choose ka, kb, kc ∈ Z \ {0} such
that kamana + kb
mb
nb
+ kc
mc
nc
= 0 and that ka and kb are odd while kc is even. Then on one
hand,
sgn(ωkaa−1ω
kb
b−1ω
kc
c−1) = sgn(ω
ka
a−1ω
kb
b−1ω
kc
c−1
Ka,b
)
= sgn(ωkaa−1ω
kb
b−1ω
kc
c−1
Ka,b
ω
−kaKb,a
b−1 ω
kaKb,a
b−1 )
= sgn(αkaa,bω
ℓb
b−1ω
ℓc
c−1)
with
ℓb = kbKa,b + kaKb,a,
ℓc = kcKa,b.
As v(ωℓbb−1ω
ℓc
c−1) = 0, (ℓb, ℓc) = ℓ(Kb,c,−Kc,b) for some ℓ ∈ Z with −ℓKc,b =
ℓc = kcKa,b. So we can conclude
sgn(ωkaa−1ω
kb
b−1ω
kc
c−1) = sgn(α
ka
a,bα
ℓ
b,c).
242 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
On the other hand, we see by analogous computations that
sgn(ωkaa−1ω
kb
b−1ω
kc
c−1) = sgn(ω
ka
a−1ω
kb
b−1ω
kc
c−1
Ka,c
)
= sgn(ωkaa−1ω
kb
b−1ω
kc
c−1
Ka,c
ω
−kaKc,a
c−1 ω
kaKc,a
c−1 )
= sgn(αkaa,cω
ℓ′b
b−1ω
ℓ′c
c−1) = sgn(α
ka
a,cα
ℓ′
b,c)
for some ℓ′b, ℓ
′
c, ℓ
′ ∈ Z with
ℓ′b = kbKa,c = ℓ
′Kb,c,
ℓ′c = kcKa,c + kaKc,a.
We have chosen ka, kb odd and kc even. In this case, the greatest power of two that divides
kc is 2hc−ha+1. On the other hand, the greatest power of two that divides Kc,a and Kc,b is
2hc−ha . We can thus conclude from ℓKc,b = −kcKa,b and ℓ′Kb,c = kbKa,c that ℓ is even
while ℓ′ is odd since Ka,b,Ka,c and Kb,c are all odd. So we see that
sgn(ωkaa−1ω
kb
b−1ω
kc
c−1) = sgn(αa,b) = sgn(αa,cαb,c),
which proves the necessity of the condition.
Now suppose sgn(αa,bαa,cαb,c) = 1 for all a, b, c ≥ 0 with ha = hb ≤ hc. Let
K := (k0, . . . , kr) ∈ Zr+1 be such that
∑r
i=0 ki
mi
ni
= 0. Let suppK := {i | ki ̸= 0}
and n := | suppK|. We prove that Πri=0ω
ki
i−1 ∈ R is uniquely determined by induction on
n ≥ 2. We first suppose 0 ̸∈ suppK. We will deal with the case 0 ∈ suppK at the end of
our proof.
If n = 2, then Πri=0ω
ki
i−1 = ω
ki
i−1ω
kj
j−1 for some i, j > 0, and its value in the residue
field is a power of αi,j .
Now suppose n > 2. Take two distinct a, b ∈ suppK. As at least one of Ka,b and
Kb,a is odd, so suppose Ka,b is odd. Then
Πri=1ω
ki
i−1
Ka,b
= Πri=1ω
ki
i−1
Ka,b
ω
−kaKb,a
b−1 ω
kaKb,a
b−1 = Π
r
i=1ω
ℓi,1
i−1α
ka
a,b
with ℓa,1 = 0, ℓb,1 = kaKa,b + kbKb,a and ℓi,1 = kiKa,b for i ̸= a, b. Since |{i | ℓi,1 ̸=
0}| is strictly smaller than n, Πri=0ω
ℓi,1
i−1 ∈ R is uniquely determined by the induction
hypothesis. So we have determined Πri=0ω
ki
i−1
Ka,b
∈ R. As Ka,b is odd, Πri=0ω
ki
i−1 ∈ R is
determined as well.
We now need to show that in this way, Πri=1ω
ki
i−1 is uniquely determined, that is, if we
choose another a′, b′ ∈ suppK instead of a, b, we get the same value for Πri=1ω
ki
i−1 ∈ R.
We will show this by choosing c ∈ suppK \ {a, b} and proving that the evaluated value of
Πri=1ω
ki
i−1 is the same whether we factor a power of αa,b as above, or αa,c or αb,c instead.
By transitivity of the equality relation, this will imply that the obtained value of Πri=1ω
ki
i−1
is independent of the choice of a, b ∈ suppK. Suppose without loss of generality that
ha ≤ hb ≤ hc and that Ka,b,Ka,c and Kb,c are odd. Above, we have evaluated
Πri=1ω
ki
i−1
Ka,b
= Πri=1ω
ℓi,1
i−1α
ka
a,b
L. Vukšić: Valuations and orderings on the real Weyl algebra 243
with ℓa,1 = 0, ℓb,1 = kaKa,b + kbKb,a and ℓi,1 = kiKa,b for i ̸= a, b. We proceed by
evaluating, in the same way as before,
Πri=1ω
ℓi,1
i−1
Kb,c
= Πri=1ω
pi,1
i−1α
ℓb,1
b,c
with pa,1 = pb,1 = 0, pc,1 = ℓc,1Kb,c + ℓb,1Kc,b and pi,1 = ℓi,1Kb,c for i ̸= a, b, c. So
Πri=1ω
ki
i−1
Ka,bKb,c
= Πri=1ω
pi,1
i−1α
kaKb,c
a,b α
ℓb,1
b,c (3.2)
with ℓi,1 and pi,1 for all 0 ≤ i ≤ r as above. In particular, we see that for i ̸= a, b, c,
pi,1 = ℓi,1Kb,c = kiKa,bKb,c. Similarly, we can compute
Πri=1ω
ki
i−1
Ka,cKb,c
= (Πri=1ω
ℓi,2
i−1α
ka
a,c)
Kb,c = Πri=1ω
pi,2
i−1α
kaKb,c
a,c α
ℓb,2
b,c (3.3)
with
1. ℓa,2 = 0, ℓc,2 = kaKa,c + kcKc,a ℓi,1 = kiKa,c for i ̸= a, c, and
2. pa,2 = pb,2 = 0, pc,2 = ℓb,2Kc,b + ℓc,2Kb,c, pi,2 = kiKa,cKb,c for i ̸= a, b, c.
Let N := Ka,bKa,cKb,c. On one hand, we see from 3.2 that
Πri=1ω
ki
i−1
N
= Πri=1ω
pi,1
i−1
Ka,c
α
kaKb,cKa,c
a,b α
ℓb,1Ka,c
b,c , (3.4)
and on the other hand, we see from 3.3 that
Πri=1ω
ki
i−1
N
= Πri=1ω
pi,2
i−1
Ka,b
α
kaKb,cKa,b
a,c α
ℓb,2Ka,b
b,c . (3.5)
We need to show that in both equations, we get the same value. We first see that for all
i ̸= c, pi,1Ka,c = pi,2Ka,b. So, given that
r∑
i=1
pi,1
mi
ni
=
r∑
i=1
pi,2
mi
ni
= 0,
we can see pc,1Ka,c = pc,2Ka,b holds as well, and thus we conclude
Πri=1ω
pi,1
i−1
Ka,c
= Πri=1ω
pi,2
i−1
Ka,b
.
It then follows that
α
kaKa,cKb,c
a,b α
ℓb,1Ka,c
b,c = α
kaKb,cKa,b
a,c α
ℓb,2Ka,b
b,c ,
since both sides of the equation are equal to ωNkaa−1 ω
Nkb
b−1 ω
Nkc−pc,1Ka,c
c−1 and the signs of
αa,b, αa,c and αb,c were chosen so that the signs of both sides of the equality match. We
conclude that the value of Πri=1ω
ki
i−1
N
is the same in both 3.4 and 3.5. As N is odd (since
Ka,b, Ka,c and Kb,c are all odd), we conclude that Πri=1ω
ki
i−1 is the same whether we factor
a power of αa,b or αa,c. If we factored a power of αb,c, we would, as similar computations
as above would show, get the same value for Πri=1ω
ki
i−1.
244 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
We have now shown that if the condition of the proposition is fulfilled, Πri=1ω
ki
i−1 is
uniquely determined for all k1, . . . , kr ∈ Z with
∑r
i=1 ki
mi
ni
= 0.
Now we consider the case 0 ∈ suppK. Let K = (k0, . . . , kr) ∈ Zr+1 be such that
k0 ̸= 0 and
∑r
i=0 ki
mi
ni
= 0. Let N := gcd{ni | i ∈ suppK}. If N is odd, i.e., if ni is
odd for every i ∈ suppK, then Πri=0ω
ki
i−1 is uniquely determined. This is because
Πri=0ω
ki
i−1
N
= (Πri=0x
ki
mi
ni Πri=0ω
ki
i−1)
N = Πi=1(xmiωnii−1)
kici = Πri=1β
kici
i
where ci := Nni for each i ≤ i ≤ r. We thus conclude Π
r
i=0ω
ki
i−1 ∈ R is the uniquely
determined N -the real root of Πri=1β
kici
i . Now suppose nj is even for some j ∈ suppK.
Then mj must be odd since gcd(mj , nj) = 1. Let k′j := kj − k0nj and k′i := ki for all
i ∈ suppK \ {0, j}. Then
∑r
i=1 k
′
i
mi
ni
= 0 and
Πri=0ω
ki
i−1
mj
= (xmjωnj )k0(Πri=1ω
k′i
i−1)
mj = βk0j (Π
r
i=1ω
k′i
i−1)
mj .
We evaluate (Πri=1ω
k′i
i−1)
mj as above k0 = 0 and conclude that Πri=0ω
ki
i−1 ∈ R is the
unique mj-th real root of βk0j (Π
r
i=1ω
k′i
i−1)
mj .
This concludes the proof of our proposition.
In Lemma 3.6, we suppose that v is a valuation on D1(R) extended from (ωi)i≥−1 to
D1(R) and compute the value of certain elements of D1(R) in this case.
Lemma 3.5. Let D be a division ring endowed with a valuation v with an abelian value
group and a commutative residue field with characteristic zero. Let a, b ∈ D be such that
a ∼ b, v(a) = v(b) = 0 and v(ab) < v[a, b]. Then v(an − bn) = v(a − b) for all
n ∈ Z \ {0}. If there exist c, d ∈ D such that cn = a, dn = b, c = d for some n ∈ N, then
v(cm − dm) = v(a− b) for all m ∈ Z.
Proof. For n ∈ N, write an − bn = (a− b)
∑n−1
i=0 a
n−1−ibi + terms with higher v-value.
Since an−1−ibi is the same for all 0 ≤ i ≤ n − 1, the v-value of the sum is equal to
zero, proving the statement for positive integers n. For negative n ∈ Z, the statement
follows from an − bn = −an(a−n − b−n)bn. The last statement of the lemma follows
from a− b ∼ (c− d)
∑n−1
i=0 c
n−1−ibi.
Lemma 3.6. Suppose v is a valuation on A1(R) and suppose i1, i2, . . . , ir ∈ N and
k0 ∈ Z, ki1 , ki2 , . . . , kir ∈ Z \ {0} are such that v(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1) = 0. If
min1≤j≤r{v(ωij )} is achieved at exactly one j, then
v(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1 − xk0ω
ki1
i1−1 · · ·ω
kir
ir−1) = min{v(ωij ) | 1 ≤ j ≤ r}.
Proof. Let n be the least common multiple of ni1 , . . . , nir and cij =
n
nij
for each ij . Since
(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1)
n = Πrj=1x
nkij
mij
nij (ω
ki1
i1−1 · · ·ω
kir
ir−1)
n
= Πrj=1(x
mijω
nij
ij−1)
kij cij = Πrj=1β
kij cij
ij
,
L. Vukšić: Valuations and orderings on the real Weyl algebra 245
by Proposition 2.2, we can compute
(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1)
n −Πrj=1β
kij cij
ij
∼
r∑
j=1
(xmi1ω
ni1
i1−1)
ki1ci1 · · · (xmij−1ω
nij−1
ij−1−1)
kij−1cij−1 ·
((xmijω
nij
ij−1)
kij cij − β
kij cij
ij
)β
kij cij
ij+1
· · ·β
kij cij
ir
.
For each j such that kij > 0,
(xmijω
nij
ij−1)
kij cij − β
kij cij
ij
∼ ωij
kij cij−1∑
i=0
(xmijω
nij
ij−1)
kij cij−iβiij ,
which gives us v((xmijω
nij
ij−1)
kij cij − β
kij cij
ij
) = v(ωij ). In case kij < 0, we see that
(xmijω
nij
ij−1)
kij cij − β
kij cij
ij
=
− (xmijω
nij
ij−1)
kij cij β
kij cij
ij
((xmijω
nij
ij−1)
−kij cij − β
−kij cij
ij
),
which again implies v((xmijω
nij
ij−1)
kij cij − β
kij cij
ij
) = v(ωij ). We conclude, using
Lemma 3.5,
v(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1 − xk0ω
ki1
i1−1 · · ·ω
kir
ir−1)
= v((xk0ω
ki1
i1−1 · · ·ω
kir
ir−1)
n − (xk0ωki1i1−1 · · ·ω
kir
ir−1)
n)
= min{v(ωij ) | 1 ≤ j ≤ r}.
With the help of Lemma 3.6, we will evaluate v(xk0ωki1i1−1 · · ·ω
kir
ir−1 −
xk0ω
ki1
i1−1 · · ·ω
kir
ir−1) when v(x
k0ω
ki1
i1−1 · · ·ω
kir
ir−1) = 0 in general. As in Lemma 3.6, we
assume k0 ∈ Z, kij ∈ Z \ {0} for all 1 ≤ j ≤ r. This will be helpful when we will
later construct a valuation v associated to a sequence (ωi)i≥−1. Let us assume for now that
i1 < i2 < · · · < ir; at the end of the calculation we will see that the order of ij does not
affect the v-value.
To start, we introduce some abbreviations to make the written equations easier to read.
Let n and cij for all j be as in the proof of Lemma 3.6,
A0 := x
k0ω
ki1
i1−1 · · ·ω
kir
ir−1
B0 := A0
A
(n)
0 := x
nk0ω
nki1
i1−1 · · ·ω
nkir
ir−1
B
(n)
0 := A
(n)
0 = Π
r
j=1β
kij cij
ij
.
Since B0 is in R, we can write
A0 −B0 = (An0 −Bn0 )(
n−1∑
i=0
An−i−10 B
i
0)
−1 ∼ (A(n)0 −B
(n)
0 )(
n−1∑
i=0
An−i−10 B
i
0)
−1. (3.6)
246 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
Since v(
∑n−1
i=0 A
n−i−1
0 B
i
0) = v(A
n−i−1
0 B
i
0) = 0 for all i, which holds due to A
n−i−1
0 B
i
0
= Bn−10 for all i, v(A0 −B0) = v(A
(n)
0 −B
(n)
0 ). To evaluate the right-hand side of (3.6),
we first proceed as we have done in the proof of Lemma 3.6, so
A
(n)
0 −B
(n)
0 ∼
r∑
j=1
Πj−1ℓ=1(x
miℓω
niℓ
iℓ−1)
kiℓciℓ · ((xmijω
nij
ij−1)
kij cij − β
kij cij
ij
)·
Πrℓ=j+1β
kij cij
iℓ
.
If kij > 0, we proceed by
(xmijω
nij
ij−1)
kij cij − β
kij cij
ij
∼ ωij
kij cij−1∑
p=0
(xmijω
nij
ij−1)
kij cij−p−1βpij .
For ij > 0, kij < 0, we can on the other hand write
(xmijω
nij
ij−1)
kij cij − β
kij cij
ij
∼ −ωij (x
mijω
nij
ij−1)
kij cij β
kij cij
ij
−kij cij−1∑
p=0
(xmijω
nij
ij−1)
−kij cij−p−1βpij .
We now define, if kj > 0,
Cj :=
kij cij−1∑
p=0
Πj−1ℓ=1(x
miℓω
niℓ
iℓ−1)
kiℓciℓ · (xmijω
nij
ij−1)
kij cij−p−1βpij ·Π
r
ℓ=j+1β
kiℓciℓ
iℓ
,
and, if kj < 0,
Cj := −(xmijω
nij
ij−1)
kij cij β
kij cij
ij
−kij cij−1∑
p=0
Πj−1ℓ=1(x
miℓω
niℓ
iℓ−1)
kiℓciℓ ·
(xmijω
nij
ij−1)
−kij cij−p−1βpij ·Π
r
ℓ=j+1β
kiℓciℓ
iℓ
for each 1 ≤ j ≤ r. Further, we define
A1,j = ωijCj
A
(n)
0 −B
(n)
0 ∼
r∑
j=1
ωijCj =
r∑
j=1
A1,j
for each 1 ≤ j ≤ r. Thus we can conclude that v(A1,j) = v(ωij ), since the image of
Cj in the residue field is equal kijcij · Πrℓ=1β
kiℓciℓ
iℓ
̸= 0 if kij > 0 and −β
2kij cij
ij
|kijcij | ·
Πrℓ=1β
kiℓciℓ
iℓ
̸= 0 if kij < 0 , making v(Cj) = 0. We can now write
A
(n)
0 −B
(n)
0 =
r∑
j=1
A1,j +A =
∑
v(A1,j) is minimal
A1,j +
∑
v(A1,j) is not minimal
A1,j .
L. Vukšić: Valuations and orderings on the real Weyl algebra 247
Here we note that the second of both finite sums on the right-hand side of this equation
includes A, which denotes the sum of all terms obtained by changing the order of factors
of the form ωiℓ (which was not explicitly written above). The fact that the v-value of these
terms is higher than the v-value of the terms of the first sum (the ones with minimal v-value)
follows from Lemma 3.2.
If min1≤j≤r{v(ωij )} is achieved at more than one j, we take the sum of all ωijCj
that have the minimal v-value, i.e.,
∑
v(A1,j) is minimal A1,j , then factor ωi1 , so the sum now
looks like∑
v(A1,j) is minimal
A1,j = ωi1
∑
v(A1,j) is minimal
ω−1i1 A1,j = ωi1
∑
v(A1,j) is minimal
ω−1i1 ωijCj
and, since v(ω−1i1 ωijCj) = 0, for each j, we can evaluate the sum of their images in the
residue field. If this sum is not equal to zero, then v(A0 − B0) = min1≤j≤r{v(A1,j)}.
Otherwise write
ωi1
∑
j
ω−1i1 ωijCj =
∑
j
ωi1(ω
−1
i1
ωijCj − ω−1i1 ωijCj).
For every j, we write ω−1i1 ωijCj − ω
−1
i1
ωijCj as an R-linear sum of terms of the form
Πℓω
nℓ
iℓ−1 (in the same way we did with A0 −B0). We sum all of the newly obtained terms,
as well as the terms in
∑
v(A1,j) is not minimal A1,j , and relabel them as A2,j where j goes
from 1 to the number of all terms.
As A0 −B0 can be written in the form
A0 −B0 = (
∑
v(A2,j) is minimal
A2,j +
∑
v(A2,j) is not minimal
A2,j)D,
where we use D as the label of the product of all terms of the form (
∑
i A
k−iBi)−1 where
A = xk
′
0Πrj=1ω
k′ij
ij−1 for some i1, . . . , ir ∈ N, k
′
0, k
′
i1
, . . . , k′ir ∈ Z and B = A, that we
factor out when we evaluate ω−1i1 ωijCj −ω
−1
i1
ωijCj for each j. All terms (
∑
i A
k−iBi)−1
have v-value equal to zero and their image in the residue field, which is of the form
(mBm−1)−1 ∈ R for some m ∈ N, is easy to determine.
We repeat the described procedure, writing A0 − B0 = (
∑
j Ak,j)D for increasing k.
We stop when for some k,
∑
j,v(Ak,j) is minimal Ak,j is either composed of one single term
or, after factoring out one of the terms, the image of the sum in the residue field is not zero.
In this case, we conclude that v(A0 − B0) is equal to v(Ak,j) for any term of the sum∑
j,v(Ak,j) is minimal Ak,j .
We must show that the process ends at some point even if the number of terms whose v-
value we evaluate at each step is growing. We see that whenever we write xk0Πrℓ=1ω
kℓ
iℓ−1−
xk0Πrℓ=1ω
kℓ
iℓ−1 as a sum of terms with strictly positive v-value, the value of each of these
terms is v(ωiℓ) for some ℓ = 1, . . . , r. It follows that v(A0 − A0) is a sum of v(ωℓ) for
some ℓ ≥ 1.
If v(ωN ) is irrational for some N ≥ 0, the process either stops beforehand or, after
k ≥ N − ir steps we get a unique term Ak,j that has v-value equal to v(ωirωir+1 · · ·ωN ).
This is the term we get when we take the last term of A(n)0 − B
(n)
0 , written as a sum of
terms A1,j with higher v-value and in each of the following steps whenever the v-value of
248 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
this term is minimal, take the last term when Ak,j is written as a sum of terms with higher
v-value.
If on the other hand, v(ωk) ∈ Q for an infinite sequence (ωk)k≥−1, then
limk→∞ v(ωk) = 0 since by Lemma 3.2,
∑k
i≥−1 v(ωk) < 0 for all k ≥ −1 and v(ωi) > 0
for i ≥ 1. Then for some N ≥ 1, v(ωN ) < v(ωi) for all 1 ≤ i < N . The evaluation of
v(A0 − B0) again either stops beforehand or we get a unique term Ak,j that has v-value
equal to v(ωirωir+1 · · ·ωN ). As in the first case, this term is the one we get when we take
the last term of A(n)0 − B
(n)
0 , written as a sum of terms A1,j with higher v-value and in
each of the following steps whenever the v-value of this term is minimal, take the last term
when Ak,j is written as a sum of terms with higher v-value.
In both cases, the value of this term, v(ωirωir+1 · · ·ωN ) is strictly smaller than the
value of all additional terms we get when we change the order of the factors in a product.
It follows that given the v-values of (ωi)i≥−1, v(A0 − B0) is the same as it would be if
all elements of the sequence (ωi)i≥−1 commuted. This follows from the fact that when we
change the order of factors ωi and ωj in some Ak,j1 , the term we obtain has v-value greater
by v[ωi, ωj ] − v(ωiωj) = −v(xyω1 · · ·ωj) for −1 ≤ i < j ≤ N while v(Ak+1,j1) −
v(Ak,j2) is for any j1, j2 equal to v(ωiℓ+1) for some iℓ such that Ak,j2 contains a power
of ωiℓ . Since, as we have shown in the proof of Lemma 3.2,
∑N
i=−1 v(ωi) < 0, it follows
that the terms we obtain by changing the order of the factors have v-value greater than
v(A0 − B0). And since v(xk0ω
ki1
i1−1 · · ·ω
kir
ir−1 − xk0ω
ki1
i1−1 · · ·ω
kir
ir−1) is higher than the
v-value of any terms we get when we change the order of factors, the order of ωi1 , . . . , ωir
does not matter.
3.2 Extending v from the sequence (ωi)i≥−1 to A1(R)
We can now prove that every v associated to either a finite or an infinite sequence (ωi)i≥−1
can be extended to a valuation on D1(R).
Lemma 3.7. For every r ≥ 0, there exists a finite number M of elements of the form
ai := ω
ki,−1
−1 ω
ki,0
0 ω
ki,1
1 · · ·ω
ki,r−1
r−1 for some ki,1, . . . , ki,r−1 ∈ Z such that v(ai) = 0 for
all 1 ≤ i ≤ M and every ωℓ−1−1 · · ·ω
ℓr−1
r−1 ∈ D1(R) with v-value zero is ∼-equivalent to a
product of positive integer powers of a1, . . . , aM .
Proof. Since v(ω−1) = −1 and v(ωj) = mj+1nj+1 ∈ Q for j ≥ 0, the problem translates to
finding general classes of solutions to the diophantine equation x0a0 + · · ·xkak = 0 with
a0 = −Πkj=1ni and ai = mini Π
k
j=1ni for all 0 ≤ i ≤ k.
Theorem 3.8. Let v and (ωi)i≥−1 be as described in the beginning of the section, i.e.,
ω−1 = x, ω0 = y, v(ω−1) = −1, v(ωi) = mi+1ni+1 ∈ Q, x
mi+1ω
ni+1
i = βi+1 ∈ R,
ωi+1 = x
mi+1ω
ni+1
i − βi+1 for i, 0 ≤ i ≤ N − 1 and v(ωN ) ̸∈ Q for some N ≥ 0 or
v(ωi) ∈ Q for infinitely many i, that
∑k
i=−1 v(ωi) < 0 for all k ≥ −1. Suposse that sgnβi
is constant on the set of all i for which ni is even. Then v can be extended to a valuation
on D1(R) with residue field R. The valuation is unique for every choice of {αi,j}i,j≥0
where αi,j = ω
Ki,j
i−1 ω
Kj,i
j−1 ,Ki,j =
mjni
di,j
,Kj,i = −minjdi,j with di,j = gcd{mjni,minj}.
The associated value group is group-isomorphic to a subgroup of Q × Z generated by
{v(ωi)}i≥−1.
L. Vukšić: Valuations and orderings on the real Weyl algebra 249
Proof. The following construction of the valuation v associated to the sequence (ωi)i≥−1
was first sketched in [22]. Here we present it in full detail.
Before we begin with the construction of the v-value for an arbitrary element of D1(R),
we define it for some specific elements of D1(R).
1. Since we have defined v(ωi) for all −1 ≤ i ≤ N , v(ΠNi=−1ω
ki
i ) =
∑N
i=−1 kiv(ωi)
must hold for all k−1, . . . , kN ∈ Z.
2. Since we supposed
∑k
i=−1 v(ωi) < 0 for all k ≥ −1, it follows from Lemma 3.2
that
v[ωi, ωj ] = −v(ω−1ω0 · · ·ωi−1ωi+1 · · ·ωj−1),
which must be strictly greater than v(ωiωj) for all i < j.
3. We also see that if v(ωk0−1 · · ·ω
kr
r−1) = 0 for some k0, . . . , kr ∈ Z, then ω
k0
−1 · · ·ω
kr
r−1
is uniquely determined by {αi,j}i,j≥0 as in shown in Proposition 3.4. In this case,
v(ωk0−1 · · ·ω
kr
r−1−ω
k0
−1 · · ·ω
kr
r−1) must be equal to the value determined in Lemma 3.6
and the discussion following it.
In all three cases, the chosen values were the only possible extensions of v from (ωi)i≥−1
if we want v to be a valuation.
To determine v(F ) for any F ∈ A1(R), we first note that F can be written as a finite
sum
F =
∑
ℓ
αℓx
iℓyjℓ , αℓ ∈ R.
Let F1 be the sum of all terms αℓxiℓyjℓ such that iℓv(x) + jℓv(y) is equal to u :=
minℓ{iℓv(x) + jℓv(y)}.
If F1 consists of only one such term, then we define v(F ) = u; this is obviously always
the case whenever v(y) ̸∈ Q. Otherwise, we factor out xi1yj1 with the smallest power of x
and get
F1 ∼ xi1yj1
∑
ℓ, iℓv(x)+jℓv(y)=u
αℓx
iℓ−i1yjℓ−j1 .
Since
(iℓ − i1)v(x) + (jℓ − j1)v(y) = 0,
for each ℓ in the sum, iℓ−i1jℓ−j1 = Kℓ
m1
n1
for some Kℓ ∈ Z. We can write
F1 ∼ xi1yj1f(xm1yn1)
where f(t) is a polynomial in R[t]. Since we know v(ω1) > 0 and v(α) = 0 for α ∈ R∗, v
is uniquely determined on R[ω1]. From this, it follows that v(f(xm1yn1)) = 0 if and only
if f(β1) ̸= 0 since xm1yn1 = ω1 + β1.
In this case, v(F1) = u and since all terms in F − F1 have v-value strictly greater than
u, v(F ) = u must hold. Since v(u) is a sum of integer powers of v(x) and v(y), v(F ) is in
250 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
the abelian group, generated by {v(ωi)}i≥−1. If u = 0, F = βk1f(β1) ∈ R. If on the other
hand f(β1) = 0, write f(t) = g1(t)(t− β1)k1 with g1(β1) ̸= 0 and we have
F1 ∼ xi1yj1g1(xm1yn1)ωk11 .
We set v(F1) = u + k1v(ω1) and add all terms we get from exchanging the order of
factors, whose v-value can be lower than the newly set v(F1), although still strictly higher
than u due to v[x, y] > v(xy), to F − F1. It is immediate that v(F1) is in the subgroup
of Γ, generated by v(x), v(y) and v(ω1) and that if v(F1) = 0, F1 ∈ R. It is important
to note that in both cases, we consider F1 as a single term. It follows that during our
transformation, the number of terms (if we ignore the ones we got when we changed the
order of factors in a product) is strictly smaller than before (unless, of course, F1 was just
a single term in the beginning and we get v(F ) = v(F1)).
We now consider the values of the terms in F − F1. If all of them have v-value strictly
greater than that of F1, we conclude v(F ) = v(F1). Otherwise, we take all terms of
F − F1 =
∑
ℓ′, iℓ′v(x)+jℓ′v(y)>u
αℓ′x
iℓ′ yjℓ′ , αℓ′ ∈ R
for which iℓ′v(x) + jℓ′v(y) = u′ := minℓ′{iℓ′v(x) + jℓ′v(y)} and then as before define
F2 = x
i2yj2
∑
ℓ′, iℓ′v(x)+jℓ′v(y)=u
′
αℓ′x
iℓ′−i2yjℓ′−j2 .
As above, we write F2 ∼ xi2yj2g2(xm1yn1)(x − β1)k2 , k2 ≥ 0, g2(β1) ̸= 0 and add all
the terms we get when we change the order of factors to F − F1 − F2. Their v-value is
strictly greater than u′ due to v[x, y] > v(xy).
We continue this process, defining F1, F2, . . . , Fk until all terms in F − F1 − · · · − Fk
have v-value strictly greater than min{F1, . . . , Fk}. Note that it is possible that Fk consists
of only one term from F − F1 − · · · − Fk−1.
Afterwards, we sum together all those Fi for 1 ≤ i ≤ k for which v(Fi) = u1 :=
min{v(F1), . . . , v(Fk)}. If the minimum is achieved at exactly one such Fi, we set v(F ) =
v(Fi). This is always the case whenever v(ω1) ̸∈ Q. Otherwise we can relabel the terms so
the minimum is achieved at F1, . . . , Fr for some r ≤ k. As we have shown, each Fi can be
written as Fi = xiiyjigi(xm1yn1)ωki1 . We sum the terms together, factor out x
i1yj1ωk11 ,
the term that has, written as a polynomial in x and y, the lowest power of x, and label the
new sum F1,1.
To evaluate v(F1,1), we follow a procedure similar to the one evaluating v(F1). After
factoring xi1yj1ωk11 , we are left with
F1,1 ∼ xi1yj1ωk11 (g1(xm1yn1) + g2(xm1yn1)xi2−i1yj2−j1ω
k2−k1
1 + · · ·
+ gr(x
m1yn1)xir−i1yjr−j1ωkr−k11 )
∼ xi1yj1ωk11
R∑
J=1
αJx
iJ yjJωkJ1 , with αJ ∈ R, iJ , jJ , kj ∈ Z, R ≥ 1.
Each term in the sum has v-value zero. Let a1, a2, · · · , aℓ be the terms such that each
product of the form xiyjωk1 , i, j, k ∈ Z that fulfills the condition v(xiyjωk1 ) = 0 is a
L. Vukšić: Valuations and orderings on the real Weyl algebra 251
product of positive integer powers of some of ai up to the order of factors x, y and ω1. The
existence of a1, . . . , aℓ is assured by Lemma 3.7. We can then write
F1,1 ∼ xi1yj1ωk11
R∑
J=1
γJa
m1,J
1 a
m2,J
2 · · · a
mℓ,J
ℓ
∼ xi1yj1ωk11 g(a1, · · · , aℓ), g ∈ R[t1, · · · , tℓ] (3.7)
with γJ ∈ R, mI,J ∈ Z for all 1 ≤ I ≤ ℓ and 1 ≤ J ≤ R. As before, we add all terms
we get when we change the order of multiplication of x, y or ω1 in a product to F − F1,1
since the value of its terms is strictly greater than u1. Since, as we have determined in the
beginning, each term in the sum (3.7) has v-value equal to zero and we know what ai ∈ R
is for each 1 ≤ i ≤ ℓ, v(g(a1, . . . , aℓ)) will have to be greater than or equal to zero, we can
define g(a1, . . . , aℓ) = g(a1, . . . , aℓ).
If g(a1, . . . , aℓ) ̸= 0, then we set v(g(a1, . . . , aℓ)) = 0 and v(F ) = v(F1,1) =
i1v(x) + j1v(y) + k1v(ω1). Otherwise write
g(t1, . . . , tℓ) = h(t1 − a1, . . . , tℓ − aℓ) =
L∑
i=1
Πℓj=1(tj − aj)mi,jhi(t1, . . . , tℓ)
with h, h1, . . . , hL ∈ R[t1, . . . , tn] and hi(a1, . . . , aℓ) ̸= 0 for all i. We factor out the
Πj(tj − aj)mi,j for those i for which
∑
j v(aj − aj)mi,j is minimal. Then
g(t1, . . . , tℓ) = Πj(tj − aj)mi,j g̃(t1, . . . , tk), with g̃ ∈ R[t1, . . . , tℓ].
If g̃(a1, . . . , ak) ̸= 0, we set
v(F1,1) = v(x
i1yj1) +
∑
j
mi,jv(aj − aj).
If on the other hand, g̃(a1, . . . , ak) = 0, we do the same thing as we did with g. The
process cannot go on indefinitely since g is a polynomial and hence of finite degree. All
terms we get when we exchange the order of x, y and ω1 are added to F − F1,1. Their
v-value must be strictly greater than u1. It follows from the construction that v(F1,1) must
be in the group generated by {v(ωi)}i≥−1 since this holds for v(ai − ai) for all i and that
if v(F1,1) = 0, F1,1 ∈ R.
Since v(ai − ai) is, as we have shown in Lemma 3.6 and the discussion following it, a
sum of v(ωj) and thus v(Πjω−1j (ai − ai)) = 0, we can write F1,1 as one term of the form
Πni=−1ω
ki
i g(a1, a2, . . . , aℓ) with n ∈ N and g ∈ R[t1, . . . , tℓ] and g(a1, a2, . . . , aℓ) ̸= 0.
After v(F1,1) is set, we compare it to both v(Fi) for all Fi that are not part of F1,1
and the terms of F − F1 − · · · − Fk − F1,1. If all of these terms have v-value strictly
greater than v(F1,1), then we can set v(F ) = v(F1,1). Otherwise, we collect all terms with
minimal v-value in a sum which we label F1,2. We determine v(F1,2) in the same way we
determined F1,1 and then sum all of the remaining terms that have v-value less or equal to
min{v(F1,1), v(F1,2)} to a sum labeled F1,3.
We repeat the process until for some k, min{v(F1,1), · · · , v(F1,k)} is strictly smaller
than the v-value of any of the remaining terms.
If min{v(F1,1), · · · , v(F1,k)} is achieved at exactly one i, we set v(F ) = v(F1,i).
Otherwise we sum all the terms with the minimal v-value and label the sum F2,1. We
252 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
evaluate v(F2,1) in the same way we evaluated v(F1,1). We repeat the process, defining
Fi,j and determining its v-value in the same way as above. We point out that after v(Fi,j)
is defined, we regard Fi,j as one single term in future evaluations.
Now we must show that at one point, the process ends, i.e., that for some i, j, v(Fi,j) is
strictly smaller than the v-value of all other terms. This holds because each time we define
Fi,j for some i, j, we sum a number of different terms into one single term and because
whenever we change the order of factors in a term, the degree of x and y in the difference
is strictly smaller. This means that we eventually run out of terms. We have thus defined v
for an arbitrary polynomial F ∈ A1(R). What we essentially did was that we wrote
F = F̃ + F̃1
where F̃ is written as a single term, v(F̃ ) is computed as if x and y commuted and the
v-value of each term of F̃1 is strictly greater than v(F̃ ). For another G ∈ A1(R), we can
write
FG = F̃G+ ˜(FG)1
and since we evaluate v(F̃ ) and v(G̃) as if x and y commuted, v(F̃G) = v(F̃ ) + v(G̃).
We use the same reasoning to show v(F +G) ≥ min{v(F ), v(G)}.
It follows from the construction that for each i, j, v(Fi,j) is a linear combination of
{v(ωi)}i≥−1 and that in case v(Fi,j) = 0, Fi,j ∈ R.
Theorem 3.9. Let v be a valuation on A1(R) trivial on R with residue field R. Then v is
strongly abelian.
Proof. If v’s value group is Q, then the theorem follows from Corollary 2.4. Otherwise,
v(ωN ) ̸∈ Q for some N by our construction. But as we have shown in Lemma 3.2,
v[ωN , x] = −v(yω1 · · ·ωN−1). If v(ωNx) = v[ωN , x], it follows that v(ωN ) ∈ Q, a
contradiction. Since the value group is generated by {v(ωi)}i≥−1, it follows from Propo-
sition 2.7 that v is strongly abelian.
4 Valuations on R[y; δ]
In this section, we explain a construction of valuations on the ring R[y; δ] with
R := {
∑
k≥m
akx
− kn | ak ∈ R,m ∈ Z, n ∈ N}
and δ(p(x)) = p′(x). This construction, which was first introduced in [16], will, as we
will see in this section, give us all valuations on R[y; δ] with residue field R. Then, we
will prove exactly which valuations on A1(R) with residue field R extend to a valuation on
R[y; δ] with the same residue field, answering the question posed by Marshall and Zhang
in [16]. We will see the extensions of valuations on R[y; δ] are strongly abelian.
Every valuation on R[y; δ] can be uniquely extended to its quotient ring, which we label
as D, because R[y; δ] is an Ore domain. Since [y, x] = 1 as before, v(xy) < 0 must hold.
We set v(x) = −1, z0 := y and consider v(y). If v(y) ̸∈ Q, then
v(
n∑
i=0
pi(x)y
i) = min
0≤i≤n
{v(pi(x)) + iv(y)}
L. Vukšić: Valuations and orderings on the real Weyl algebra 253
for any
∑n
i=0 pi(x)y
i ∈ R[y; δ]. Otherwise v(y) = r1 ∈ Q and hence v(y − γ1x−r1) >
v(y) for some γ1 ∈ R. If v(z1) = r2 ∈ Q for z1 := y − γ1x−r1 , we proceed to find
z2 = z1 − γ2x−r2 such that v(z2) is greater than r2. We repeat this process to construct a
sequence (zi)i≥0.
If v(zk) ̸∈ Q for some k ∈ N, then we can write every f ∈ R[y; δ] as
∑n
i=0 pi(x)z
i
k
and deduce
v(f) = min
0≤i≤n
{v(pi(x)) + iv(zk)}.
The value group is then group-isomorphic to Q×Z. Since v[x, zk] = v[x, y] = 0 > v(xzk),
v is strongly abelian by Proposition 2.7. Otherwise, the sequence (zi)i≥0, v(zi) = ri+1 ∈
Q is infinite. We take note of the fact that v(zi+1) > v(zi) and since [zi, x] = [y, x] = 1
for all i, v(zi) < 1 for all i. We define r := limi→∞ ri ≤ 1.
4.1 Case r < 1
If r < 1, it has been shown in [16] that v can be extended to a valuation on R[y; δ] with
residue field R. We first extend v from R to
R̃ = {
∑
q∈A
aqx
−q | aq ∈ R, A ⊂ Q is well-ordered}
in a natural way, i.e., by defining
v(
∑
q∈A
aqx
−q) = minA
for each
∑
q∈A aqx
−q ∈ R̃[y; δ]. Then for every f(t) ∈ R[t], define v(f(y)) = v(f(z))
with z :=
∑∞
i≥1 aix
−ri and f(y) =
∑n
i=0 piy
i for f(t) =
∑n
i=0 pit
i. This gives rise to a
valuation on R[y; δ].
However if r = 1, we cannot define a valuation in this way. Let k ∈ N be such that
2rk > 1 + r1, which exists since r = 1, and ak = y − zk =
∑k
i=1 γix
−ri . Let
f(t) = (t− ak)(t− ak) = t2 − 2tak + a2k ∈ R[t].
On one hand,
v(f(y)) = v(f(z)) = 2v(z − ak) = 2rk+1.
On the other,
2rk+1 = v((y−ak)(y−ak)) = v(y2−2yak+a2k+[y, ak]) = min{2rk+1, 1+r1} = 1+r1,
contradicting the assumption that v is a valuation, as shown in [16].
Of course, even in case r < 1, there may also exist a k such that 2rk > 1 + r1. But the
important difference between the two cases is that if r < 1, there is always an ℓ ∈ N such
that 1 + rℓ > 2rk for all k ∈ N, which does not hold in case r = 1. Then, since
f : R[y; δ] → R[y; δ], f(y) = zℓ, f(a) = a for a ∈ R
is a real algebra automorphism of R[y; δ], we can translate the sequence by replacing y
with zℓ.
We see that since the associated value group is Q, v is a strongly abelian valuation by
Corollary 2.4.
254 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
4.2 Case r = 1
The question whether in case r = 1, v can be extended from a sequence (zi)i≥0 to a
valuation on R[y; δ] was left open in [16]. In this subsection, we show that it can be done
using model theory (for reference, see for example [19]). We also show that the valuation
we get in this way is uniquely determined.
Suppose we have infinite sequences (zi)i≥0 ⊆ R[y; δ], (ri)i≥1 ⊆ Q and (γi)i≥1 ⊆ R
and v : (zi)i≥0 → Q with z0 = y, zi+1 = zi − γi+1x−ri+1 and v(zi) = ri+1 ∈ Q with
(ri)i≥1 a strictly increasing sequence with r = limi→∞ ri = 1. Then for each n ≥ 0, there
is a valuation vn on R[y; δ] such that vn(zi) = ri+1 for all 0 ≤ i ≤ n− 1 and vn(zn) ̸∈ Q.
We now present the first-order theory that the valuation associated to the infinite se-
quence we wish to prove exists is a model of. The theory will be a union of the theory of
D, the quotient division ring of R[y; δ] and the theory of valuations. We will see that each
finite subset of this theory has a model. By compactness, so does the whole theory.
The language of our theory will be
F ∪ {+,−, ·,−1 , O,<} ∪ {czi | i ≥ −1}
where F is the set of all constants ca for each a ∈ D, +, · and < are binary function
symbols, − and −1 are unary function symbols, O is an unary relation symbol and czi is a
constant for all i ≥ 0. Let A be the theory of the quotient division ring of the ring R[y; δ].
By B we will denote the set of axioms for valuation rings O on division rings:
1. O(0) ∧O(1)
2. ∀a : O(a) ∨O(a−1)
3. ∀a, b : O(a) ∧O(b) → O(a+ b) ∧O(ab) ∧O(ba)
We add all sentences C that will give proper meaning to the constants czi for all i ≥ −1:
4. cz0 = cy
5. czi+1 = czi − cγi+1x−ri+1
6. O(cxri+1zi) ∧O(c(xri+1zi)−1)
Our theory is then the union of all the above axioms from A to C. Since all finite
subsets of the theory have a model, namely, the valuation vn described in the beginning of
this subsection, so does, by compactness, the whole theory. Since the theory contains F ,
the set of all constants ca for each a ∈ D, the models are valued division rings which all
contain D. We pick a model of the theory, a pair (D1, v), where D1 is a division ring with
valuation v.
We now show that the v-value is uniquely determined for every f ∈ R[y; δ]. It will
then follow that v is uniquely determined on the whole quotient ring D. Every f ∈ R[y; δ]
can be written as
f =
n∑
i=0
p
(0)
i (x)y
i, with p(0)i (x) ∈ R for each 0 ≤ i ≤ n.
L. Vukšić: Valuations and orderings on the real Weyl algebra 255
For the time being, we ignore the terms we get when we change the order of multipli-
cation. At the end of this subsection, we will see that they do not influence v(f). For each
k ≥ 1, we define
ak := y − zk =
k∑
i=1
γix
−ri
and write
f =
n∑
i=0
p
(0)
i (x)y
i =
n∑
i=0
p
(0)
i (x)(ak + zk)
i
∼
n∑
i=0
p
(0)
i (x)
i∑
j=0
(
i
j
)
ai−jk z
j
k =
n∑
j=0
p
(k)
j (x)z
j
k
with
p
(k)
j (x) :=
n∑
i=j
(
i
j
)
p
(0)
i (x)a
i−j
k
for each 0 ≤ j ≤ n and k ≥ 1. For each 0 ≤ j ≤ n, gj(t) :=
∑n
i=j
(
i
j
)
p
(0)
i (x)t
i is a
polynomial in R[t]. The quotient field of
C := {
∑
k≥m
akx
− kn | ak ∈ C,m ∈ Z, n ∈ N}
is the algebraic closure of the quotient field of R, as shown in for example [14]. Since∑∞
i=1 γix
−ri is not in the quotient field of C, it is not a root of gj(t) for any 0 ≤ j ≤ n.
We conclude that for some K ∈ N, v(p(k)j (x)) = v(p
(K)
j (x)) for all k ≥ K and all
0 ≤ j ≤ n. We can then write
f =
n∑
i=0
p
(0)
i (x)y
i =
n∑
i=0
p
(K)
i (x)z
i
K =
n∑
i=0
p
(k)
i (x)z
i
k
with v(p(k)i (x)) = v(p
(K)
i (x)) for all k ≥ K.
We now show that from some K ′ ≥ K, v(p(k)0 ) < v(p
(k)
i z
i
k) for all 1 ≤ i ≤ n and all
k > K ′. For all k > K,
p
(k+1)
0 (x) =
n∑
i=0
p
(k)
i (x)(ak+1 − ak)
i =
n∑
i=0
p
(k)
i (x)(γk+1x
−rk+1)i
with v(p(k)i (x)) = v(p
(K)
i (x)). Since (ri)i≥1 is an increasing sequence with limi→∞ ri =
1, there exists K ′ ≥ K such that
min
i=0,...,n
{v(p(K
′)
i (x)) + irK′+1} = min
i=0,...,n
{v(p(K)i (x)) + irK′+1}
is achieved at exactly one 0 ≤ i ≤ n. We conclude v(p(K
′+1)
0 ) = v(p
(K)
0 ). Then for each
1 ≤ i ≤ n,
v(p
(k)
i (x)z
i
k) = v(p
(k)
i (x))+irk+1 = v(p
(K)
i (x))+irk+1 > v(p
(K)
i (x))+irk ≥ v(p
(K)
0 ).
256 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
To show that v(f) is equal to v(p(K)0 (x)) ∈ Q, we must show that the v-value of all
the terms we get when we change the order of multiplication must be strictly greater than
v(p
(K)
0 (x)). For all k ≥ 1, we write
p
(k)
0 (x) =
n∑
i=0
p
(0)
i (x)a
i
k =
n∑
i=0
p
(0)
i (x)(
k∑
j=1
γjx
−rj )i =
n∑
i=0
ki∑
j=1
p
(0)
i (x)αix
−qj ,
with qj ∈ Q for all 1 ≤ j ≤ ki and 1 ≤ i ≤ n. Since for all 0 ≤ i ≤ n, p(0)i (x) ∈ R and
(ri)i≥1 is an increasing sequence with limi→∞ ri = 1, there exists some k ≥ 1 such that
the term of the sum with v-value
min
i=1,...,n
{v(p(0)i (x)) + (i− 1)r1 + rk}
is the only term in the sum with its v-value. We conclude
v(p
(K)
0 (x)) = v(p
(k)
0 (x)) ≤ v(p
(0)
i (x)) + (i− 1)r1 + rk (4.1)
for all 1 ≤ i ≤ n. On the other hand, we can write
f =
n∑
i=0
p
(0)
i (x)y
i =
n∑
i=0
p
(0)
i (x)(ak + zk)
i.
For all 0 ≤ i ≤ n, all terms of (ak + zk)i when expaned are of the form aℓ1k z
ℓ2
k . . . a
ℓi−1
k z
ℓi
k
with ℓ1, . . . , ℓi ≥ 0 and ℓ1 + · · ·+ ℓi = i. Since v(ak) = r1, v(zk) = rk+1 and
v[ak, zk] = v[
k∑
j=1
γjx
−rj , y −
k∑
j=1
γjx
−rj ] = v(
k∑
j=1
γj [x
−rj , y]) = 1 + r1,
it follows that the v-value of each term we get when we change the order of multiplication
is at least v(p(0)i (x)) + (i − 1)r1 + 1 for each 1 ≤ i ≤ n, which is, as is immediate from
(4.1), strictly greater than v(p(K)0 (x)). We thus conclude v(f) = v(p
(K)
0 (x)) ∈ Q. It
also follows that v(f) = vk(f) for all k ≥ K ′ with vk as defined in the beginning of this
section. As every element of D, the quotient ring for R[y; δ], can be written as fg−1 with
f, g ∈ R[y; δ], it follows that v is uniquely determined on D. We see that the value group
for v is equal to Q. We conclude from Corollary 2.4 that v is strongly abelian.
It remains to show that the residue field for v is equal to R. Suppose v(f) = 0 for some
f ∈ D. Then vk(f) = 0 for all k ≥ K for some K ∈ N. We can write
f = (
m∑
i=0
p
(K)
i z
i
K)(
n∑
j=0
q
(K)
j z
j
K)
−1
with p(K)i , q
(K)
j ∈ R and
v(
m∑
i=0
p
(K)
i z
i
K) = v(p
(K)
0 ) = v(
n∑
j=0
q
(K)
j z
j
K) = v(q
(K)
0 ) = q ∈ Q.
It follows that v(xq
∑m
i=0 p
(K)
i z
i
K) = v(x
q
∑n
j=0 q
(K)
j z
j
K) = 0 and α := xq
∑m
i=0 p
(K)
i z
i
K
= xqp
(K)
0 ∈ R = R, β := xq
∑n
j=0 q
(K)
j z
j
K = x
qq
(K)
0 ∈ R = R. We conclude
f = αβ−1 ∈ R. So the residue field for v is indeed equal to R.
L. Vukšić: Valuations and orderings on the real Weyl algebra 257
4.3 Extensions of valuations from A1(R) to R[x; δ]
In this section, we characterize the valuations on A1(R) with residue field R that have an
extension to R[y; δ] with the same residue field.
Since x
mi
ni ∈ R[y; δ], it follows that any valuation v′ that extends v to a valuation on
R[y; δ] with residue field R must satisfy v′(x
mi
ni ωi−1) = 0 and γ̃i := x
mi
ni ωi−1 ∈ R. In
the next proposition, we show the necessary condition for a valuation v on A1(R) to have
an extension v′ to R[y; δ] with the same residue field.
Proposition 4.1. Let v be a valuation on A1(R) with residue field R associated to a se-
quence (ωi)i≥−1 with v(ωi−1) = mini for i ≥ 1 and x
miωnii−1 = βi ∈ R. Let αi,j ∈ R
be as in Section 3. Let 2hi be the greatest power of two dividing ni for all i ≥ 0. Then
v can be extended to a valuation on R[y; δ] with the same residue field only if it fulfils the
following conditions:
(1) For each i such that ni is even, βi > 0 must hold, and
(2) for each i, j, ℓ with hi < hj ≤ hℓ, αi,jαi,ℓ > 0 must hold.
Proof. Since
γ̃i
ni = (x
mi
ni ωi−1)ni = xmiω
ni
i−1 = βi
due to Proposition 2.2, it is obvious that γ̃i must be equal to an ni-th root of βi. If ni is
odd, γ̃i ∈ R is uniquely determined regardless of sgn(βi), while if ni is even, γ̃i ∈ R only
if βi > 0. It is thus obvious that βi > 0 must hold for all i where ni is even if v can be
extended from a valuation on A1(R) to a valuation on R[y; δ] with the same residue field.
This proves the necessity of the first condition.
To prove the necessity of the second condition, we first observe that
αi,j = ω
Ki,j
i−1 ω
−Kj,i
j−1 = γ̃i
Ki,j γ̃j
−Kj,i ,
holds for all i, j ≥ 0. If hi < hj , Ki,j is odd while Kj,i is even, so sgn(γi) = sgn(αi,j)
must hold. We can therefore see that
αi,jαi,ℓ = γ̃i
Ki,j+Ki,ℓ γ̃j
−Kj,i γ̃ℓ
−Kℓ,i
for all i, j, ℓ ≥ 0. If hi < hj ≤ hℓ, both Ki,j and Ki,ℓ are odd while Kj,i and Kℓ,i are
even. It follows that if v can be extended to a valuation on R[y; δ] with residue field R,
αi,jαi,ℓ > 0 must hold for all i, j, ℓ ≥ 0 with hi < hj ≤ hℓ.
In this section, we show that the conditions (1) and (2) of Proposition 4.1 are also
sufficient for v to have an extension to R[y; δ] with residue field R. Let v be any valuation
on A1(R) satisfying the conditions described in Proposition 4.1. We will first determine
γ̃i ∈ R for all i ≥ 0. If ni is odd, there is a unique choice of γ̃i ∈ R. Suppose then ni
is even and βi > 0 for some i ≥ 0. If hi < hj for some j, then sgn(γ̃i) = sgn(αi,j) =
sgn(γ̃i
Ki,j γ̃j
−Kj,i) since Ki,j is even while Kj,i is odd.
We can conclude that if for every power of two 2h there is an i ≥ 0 (or, equivalently,
if the v-value group is 2-divisible), γ̃i ∈ R is uniquely determined for all i ≥ 0. If on the
other hand, the value group is non-2-divisible, there is an i ≥ 0 such that 2hi is maximal
258 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
for all i ≥ 0. We then have two choices for γ̃i - a positive or a negative one. The sign of γ̃j
is then uniquely determined for all j ≥ 0 since
αi,j = γ̃i
Ki,j γ̃j
−Kj,i ,
where Ki,j is even and Kj,i is odd. We will now take an arbitrary valuation v on A1(R)
satisfying the conditions of Proposition 4.1, constructed by a sequence of (ωi)i≥−1 as
shown in Section 3. We also pick γ̃i ∈ R for all i. Then we will describe v’s extension to
R[y; δ] with a sequence of (zi)i≥0 like in the beginning of Section 4.
Suppose the valuation v on A1(R) is given by a sequence (ωi)i≥−1 with ω−1 =
x, ω0 = y and ωi = xmiωnii−1 − βi, βi ∈ R and gcd(mi, ni) = 1 for all i. Suppose
also that v satisfies conditions (1) and (2) of Proposition 4.1. We will show that there is
exactly one extension of v from A1(R) to R[y; δ] for each appropriate choice of (γ̃i)i≥1.
Lemma 4.2. For each k ≥ ℓ ≥ 0, ωk can be written in the following form:
ωk = (Π
k
i=ℓ+1x
mi
ni )ωℓ(Π
k
i=ℓ+1Bi)−
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )γ̃i(Π
k
j=iBj)
+
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )Ai(Π
k
j=i+1Bi),
with
Ai =
ni−1∑
j=1
(x
mi
ni ωi−1)
ni−j−1x
mi
ni [x
j
mi
ni , ωi−1]ω
j
i−1
Bi =
nj∑
j=1
(x
mi
ni ωi−1)
ni−j γ̃i
j−1
for each 1 ≤ i ≤ k.
Proof. We prove the lemma by induction on k ≥ ℓ. For k = ℓ, it is trivially true since we
get ωk = ωk.
Suppose now the equation holds for some k ≥ ℓ. Then
ωk+1 = x
mk+1ω
nk+1
k − βk+1 = (x
mk+1
nk+1 ωk − γ̃k+1)Bk+1 +Ak+1
= x
mk+1
nk+1 ((Πki=ℓ+1x
mi
ni )ωℓ(Π
k
i=ℓ+1Bi)−
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )γ̃i(Π
k
j=iBj)
+
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )Ai(Π
k
j=i+1Bi))Bk+1 − γ̃k+1Bk+1 +Ak+1
= (Πk+1i=ℓ+1x
mi
ni )ωℓ(Π
k+1
i=ℓ+1Bi)−
k+1∑
i=ℓ+1
(Πk+1j=i+1x
mj
nj )γ̃i(Π
k+1
j=i Bj)
+
k+1∑
i=ℓ+1
(Πk+1j=i+1x
mj
nj )Ai(Π
k+1
j=i+1Bi),
L. Vukšić: Valuations and orderings on the real Weyl algebra 259
where we used the induction hypothesis, which is
ωk = (Π
k
i=ℓ+1x
mi
ni )ωℓ(Π
k
i=ℓ+1Bi)−
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )γ̃i(Π
k
j=iBj)
+
k∑
i=ℓ+1
(Πkj=i+1x
mj
nj )Ai(Π
k
j=i+1Bi)
in the second equation.
Since v(ωi−1) = mini and x
mi
ni ωi−1 = γ̃i, v(Bi) = 0 and Bi = niγ̃ini−1 must hold for
all i ≥ 1. From
[xmi , ωi−1] =
ni−1∑
j=0
x
mi
ni
·j
[x
mi
ni , ωi−1]x
mi
ni
·(ni−j),
we can see that v(Ai) = v[x
mi
ni , ωi−1] = v[x, ωi−1]+1− mini = −v(xyω1 . . . ωi−1). Since∑k
i≥−1 v(ωi) < 0 for every k by Lemma 3.2,
v(
k∑
i=ℓ+1
(Πj=i+1x
mj
nj )Ai(Π
k
j=iBi)) > v(ωk)
will hold for every k, which is why we can ignore the terms containing Ai during our
evaluations of v(zi) where for each i, zi ∈ R[y; δ] is as in the beginning of this section.
Lemma 4.3. Suppose v is a valuation on A1(R) constructed from a sequence (ωi)i≥−1
that extends to a valuation on R[y; δ] and thus D, its quotient division ring. For a given
i ≥ 1, define a sequence (Si,j)j≥1 by:
(a) Si,1 := (x
mi
ni ωi−1 − γ̃i)−1(Bi −Bi),
(b) Si,j+1 := (x
mi
ni ωi−1 − γ̃i)−1(Si,j − Si,j) for j ≥ 1.
Then for each j ≥ 1:
1 Si,j =
∑ni−j
k=1 Nk,j(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1 with Nk,1 = k and Nk,j+1 =∑ni−j
ℓ=k Nℓ,j , and
2 v(Si,j) = 0, Si,j = N1,j+1γ̃ini−j−1
for all 1 ≤ j ≤ ni − 1.
Proof. We prove the first statement of the lemma by induction on j ≥ 1. To show the basis
of induction, we evaluate
Bi −Bi = (x
mi
ni ωi−1 − γ̃i)(
ni∑
k=1
((x
mi
ni ωi−1)
ni−k−1+
(x
mi
ni ωi−1)
ni−k−2γ̃i + · · ·+ γ̃ini−k−1)γ̃ik−1)
= (x
mi
ni ωi−1 − γ̃i)
ni−1∑
k=1
k(x
mi
ni ωi−1)
ni−k−1γ̃i
k−1.
260 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
We can thus see Si,1 =
∑ni−1
k=1 k(x
mi
ni ωi−1)
ni−k−1γ̃i
k−1 and since x
mi
ni ωi−1 = γ̃i, we
see that the lemma holds in case j = 1. Now we suppose that the statement is true for
some j ≥ 1, i.e., Si,j =
∑ni−j
k=1 Nk,j(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1 and v(Si,j) = 0, Si,j =
N1,j+1γ̃i
ni−j−1. We then write
Si,j − Si,j =
ni−j∑
k=1
Nk,j(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1 −N1,j+1γ̃ini−j−1
=
ni−j∑
k=1
Nk,j(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1 −
ni−j∑
k=1
Nk,j γ̃i
ni−j−1
=
ni−j−1∑
k=1
Nk,j((x
mi
ni ωi−1)
ni−j−k − γ̃ini−j−1)γ̃ik−1
= (x
mi
ni ωi−1 − γ̃i)
ni−j−1∑
k=1
Nk,j((x
mi
ni ωi−1)
ni−j−1−k +· · ·+ γ̃ini−j−1−k)γ̃ik−1
= (x
mi
ni ωi−1 − γ̃i)
ni−j−1∑
k=1
(Nk,j +Nk+1,j + · · ·+Nni−j−1,j)
(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1
= (x
mi
ni ωi−1 − γ̃i)
ni−j−1∑
k=1
Nk,j+1(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1
proving the first statement of the lemma. The second statement immediately follows from
the first since x
mi
ni ωi−1 = γ̃i and thus
v(Si,j) = v(
ni−j∑
k=1
Nk,j(x
mi
ni ωi−1)
ni−j−kγ̃i
k−1) = 0,
Si,j =
ni−j∑
k=1
Nk,j(x
mi
ni ωi−1)ni−j−kγ̃i
k−1 = N1,j+1γ̃i
ni−j−1.
Lemma 4.4. Supose v is as in Lemma 4.3. For a given i ≥ 1, define a sequence (Di,j)j≥1
by:
(a) Di,1 = B1B2 · · ·Bi −B1B2 · · ·Bi,
(b) Di,j+1 = xv(Di,j)Di,j − xv(Di,j)Di,j .
Then for each j ≥ 1:
1. Di,j is a R-linear sum of terms which are products of elements from the set
{ωi}i ∪ {Bi}i ∪ {B−1i }i ∪ {Si,j}i,j , (4.2)
where parts of the product are conjugated by a rational power of x.
L. Vukšić: Valuations and orderings on the real Weyl algebra 261
2. v(Di,j) is a sum of v(ωℓ) for finitely many ωℓ.
3. xv(Di,j)Di,j is sum of products of γ̃k for various k.
Proof. Since
B1B2 · · ·Bi −B1B2 · · ·Bi =
i∑
j=1
B1 · · ·Bj−1(Bj −Bj)Bj+1 · · ·Bi
=
i∑
j=1
B1 · · ·Bj−1ωjB−1j Sj,1Bi+1 · · ·Bi,
and
x
mj+1
nj+1 B1 · · ·Bj−1ωjB−1j Sj,1Bi+1 · · ·Bi
= (x
mj+1
nj+1 B1 · · ·Bj−1x
−
mj+1
nj+1 )x
mj+1
nj+1 ωjB
−1
j Sj,1Bi+1 · · ·Bi,
for each 1 ≤ j ≤ k, the first two statements of the lemma follow from:
1. x
mi
ni ωi−1 − γ̃i = ωiB−1i ,
2. Bi −Bi = (x
mi
ni ωi−1 − γ̃)Si,1 = ωiB−1i Si,1,
3. Si,j − Si,j = (x
mi
ni ωi−1 − γ̃Si,j+1 = ωiB−1i Si,j+1,
4. B−1i −Bi
−1
= −Bi−1(Bi −Bi)Bi
−1
= −Bi−1ωiB−1i Si,1Bi
−1
,
where we ignore the terms we get when we change the order of multiplication. We can do
that that since these terms are procucts of Ai as defined in Lemma 4.2 and terms with zero
v-value. As we have already mentioned, these terms will not influence the construction
of the extension of a valuation on A1(R) to R[y; δ]. Indeed - we can see by induction on
j ≥ 1 that each term of the sum is a product of factors equal to, modulo conjugation by a
rational power of x, one of the elements of the set (4.2); that is,
1. either equal to ωi, or
2. equal to a power of Bj or Sj,i for some i, j.
Since the latter have v-value equal to zero and since both Bi − Bi and Si,j − Si,j are
products of ωi, a power of B−1i and Si,j for some i, j, v(Di,j) is a sum of v(ωi) for some
i. The last statement of the lemma follows from the fact that for all i, j, Bi and Si,j are of
the form Nγ̃i for N ∈ N.
If v is a valuation on R[y; δ] with residue field R, then, as we have presented in Sec-
tion 4, v can be constructed from a sequence (zi)i≥0 ⊂ R[y; δ]. In the next proposition, we
make the first comparison between this construction and the construction of a valuation on
A1(R) from a sequence (ωi)i≥−1 described in Section 3.
262 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
Lemma 4.5. Suppose v is as in Lemma 4.3. Suppose that for some k, ℓ ≥ 0, we can write
ωk = (Π
k
i=1x
mi
ni )zℓ(Π
k
i=1Bi) + C,
where C is a R-linear sum of elements of the form Di,j for some i, j ≥ 0. Then:
1. If v(ωk) > v(Πki=1x
mi
ni zℓ) ∈ Q, then
ωk = (Π
k
i=1x
mi
ni )zℓ+1(Π
k
i=1Bi) + C1.
2. If v(ωk) = v(Πki=1x
mi
ni zℓ) ∈ Q, then
ωk+1 = (Π
k+1
i=1 x
mi
ni )zℓ+1(Π
k+1
i=1Bi) + C2.
3. If v(ωk) < v(Πki=1x
mi
ni zℓ), then, if v(ωk) ∈ Q,
ωk+1 = (Π
k+1
i=1 x
mi
ni )zℓ(Π
k+1
i=1Bi) + C3.
Here, C1, C2 and C3 are other R-linear sums of elements of the form Di,j as in Lemma 4.5
for some i, j ≥ 0.
Proof. Suppose first v(ωk) > v(Πki=1x
mi
ni zℓ) = v(C). Since v(C) = rℓ+1 −
∑k
i=1
mi
ni
∈
Q, then rℓ+1 := v(zℓ) ∈ Q as well. So, for zℓ+1 = zℓ − x−rℓ+1γℓ+1, we can write
ωk = (Π
k
i=1x
mi
ni )zℓ+1(Π
k
i=1Bi) + γℓ+1(Π
k
i=1x
mi
ni )x−rℓ+1(Πki=1Bi) + C
= (Πki=1x
mi
ni )zℓ+1(Π
k
i=1Bi) + x
−v(C)(γℓ+1(Π
k
i=1Bi) + x
v(C)C)
Since v(zℓ+1) > v(zℓ), we see that v(γℓ+1Πki=1Bi + x
v(C)C) > 0. It follows that γi+1 =
−Πki=1Bi
−1
x−v(C)C, hence
C1 := γℓ+1Π
k
i=1Bi + x
−v(C)C = γℓ+1Π
k
i=1Bi + x
−v(C)C − x−v(C)C + x−v(C)C
= γℓ+1(Π
k
i=1Bi −Πki=1Bi) + (x−v(C)C − x−v(C)C)
We see that since C is an R-linear sum of Di,j for various i, j, so is, by Lemma 4.4,
x−v(C)C − x−v(C)C. Hence, C1 is an R-linear sum of Di,j .
Now consider the case v(ωk) = v(Πki=1x
mi
ni zℓ) ≤ v(C). Since v(ωk) = mk+1nk+1 ∈ Q,
we can evaluate
˜γk+1 = x
mk+1
nk+1 ωk = (Π
k+1
i=1 x
mi
ni )zℓ(Πki=1Bi) + x
mk+1
nk+1 C
= γℓ+1Π
k
i=1Bi + x
mk+1
nk+1 C,
L. Vukšić: Valuations and orderings on the real Weyl algebra 263
and then deduce
ωk+1 = (x
mk+1
nk+1 ωk − ˜γk+1)Bk+1
= (Πk+1i=1 x
mi
ni )zℓ(Π
k+1
i=1Bi) + x
mk+1
nk+1 CBk+1 − ˜γk+1Bk+1
= (Πk+1i=1 x
mi
ni )zℓ+1(Π
k+1
i=1Bi) + γℓ+1(Π
k+1
i=1Bi) + x
mk+1
nk+1 CBk+1 − ˜γk+1Bk+1
= (Πk+1i=1 x
mi
ni )zℓ+1(Π
k+1
i=1Bi) + γℓ+1(Π
k
i=1Bi −Πki=1Bi)Bk+1+
(x
mk+1
nk+1 C − x
mk+1
nk+1 C)Bk+1
= (Πk+1i=1 x
mi
ni )zℓ+1(Π
k+1
i=1Bi) + C2
where C2 is, again, an R-linear sum of Di,j for some i, j.
Lastly, we consider the case v(Πki=1x
mi
ni zℓ) > v(ωk) = v(C) ∈ Q. In this case,
˜γk+1 = x
mk+1
nk+1 ωk = x
mk+1
nk+1 C, so we can write
ωk+1 = (x
mk+1
nk+1 ωk − ˜γk+1)Bk+1
= (Πk+1i=1 x
mi
ni )zℓ(Π
k+1
i=1Bi) + (x
mk+1
nk+1 C − x
mk+1
nk+1 C)Bk+1
and the statement again follows.
Theorem 4.6. Suppose v is a valuation on A1(R), constructed from an either finite or
infinite sequence (ωi)i≥−1 with v(ωi) =
mi+1
ni+1
. Suppose also that v satisfies the following
conditions:
1. For each i such that ni is even, βi > 0 must hold, and
2. for each i, j, ℓ ≥ 0 with hi < hj ≤ hℓ, αi,jαi,ℓ > 0 must hold.
Then v has a unique extension to a valuation on R[y; δ] with residue field R for each choice
of (γ̃i)i≥1.
Proof. As we know from the beginning of Section 4, each valuation on R[y; δ] and D
with residue field R can be constructed by either a finite or an infinite sequence (zi)i≥0.
For every sequence (ωi)i≥−1, we will use the lemmas proved in this section to find the
unique sequence (zi)i≥0 which, as we have shown in the beginning of this section, uniquely
determines a valuation on R[y; δ]. Our calculations will then show that the valuation on D
defined by the sequence (zi)i≥0 is the extension of the valuation on A1(R) associated to
the sequence (ωi)i≥−1.
We determine the finite or infinite sequence (zi)i≥0 associated to v’s extension to
R[y; δ]. In the first step, we consider v(y). If v(y) ̸∈ Q, then v’s extension to R[y; δ]
is clearly uniquely determined, namely the one defined by
v(
n∑
i=0
pi(x)y
i) = min
0≤i≤n
{v(pi(x)) + iv(y)}
for every
∑n
i=0 pi(x)y
i ∈ R[y; δ].
264 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
So suppose v(y) = m1n1 ∈ Q. Then, in the second step of our evaluation, we write
ω1 = x
m1yn1 − β1 = (x
m1
n1 y − γ̃1)B1 = x
m1
n1 (y − x−
m1
n1 γ̃1)B1
and since v(ω1) > 0, v(y − x−
m1
n1 γ̃1) >
m1
n1
= v(y). We deduce z1 = y − x−
m1
n1 γ1 with
γ1 = γ̃1. Obviously, v(z1) = v(ω1) + m1n1 ∈ Q if and only if v(ω1) ∈ Q. If either and
hence both values are irrational, we get a unique extension of v to R[y; δ]. Otherwise, if
v(ω1) =
m2
n2
∈ Q and hence r2 = v(z2) = m1n1 +
m2
n2
, we continue with the third step of
our evaluation by writing
ω2 = (x
m2
n2 ω1 − γ̃2)B2 = (x
m2
n2 x
m1
n1 z1B1 − γ̃2)B2.
Since v(ω2) > 0, we conclude that γ2 = x
m2
n2 x
m1
n1 z1 must be equal to γ̃2B1
−1
. To evaluate
the v-value of z2 = z1 − γ2x−r2 , we write
ω2 = (x
m2
n2 x
m1
n1 z2B1 + γ2B1 − γ̃2)B2 = x
m2
n2 x
m1
n1 z2B1B2 + (γ2B1 − γ̃2)B2.
We note that ω2 is here written as a sum of Π2i=1x
mi
ni z2Π
2
i=1Bi + C where C is as in
Lemma 4.5. To determine v(z2), we compare v(ω2) and v(γ2B1 − γ̃2), the latter being
equal to v(ω1) since γ2B1 − γ̃2 = γ2(B1 −B1). There are three possible cases:
1. If v(ω2) < v(ω1), then v(x
m2
n2 x
m1
n1 z2) must be equal to v(ω2), so r3 = v(z2) is
determined by v(z2) = v(ω2) − m1n1 −
m2
n2
. If v(ω2) = m3n3 ∈ Q, then v(z2) =
r3 ∈ Q and γ3 = γ̃3(B1B2)−1 must hold. It follows that v(z2) ∈ Q if and only
if v(ω2) ∈ Q. In this case, the v-value of z3 = z2 − γ3x−r3 will be determined in
the subsequent steps, i.e., by considering (v(ωi))i≥3 and (γ̃i)i≥3. By Lemma 4.5,
ω3 = Π
3
i=1x
mi
ni z3Π
3
i=1Bi + C1, C1 being an R-linear sum of Di,j .
2. If v(ω2) = v(ω1), then v(z2) depends on γ̃3:
(a) If γ̃3 = x
m2
n2 (γ2B1 − γ̃2)B2, then v(x
m2
n2 x
m1
n1 z2) must be greater than v(ω1)
since in this case, ω2 ∼ (γ2B1 − γ̃2)B2 and will be determined in the sub-
sequent steps, i.e., by considering (v(ωi))i≥3 and (γ̃i)i≥3. By Lemma 4.5,
ω3 = Π
3
i=1x
mi
ni z2Π
3
i=1Bi + C2, C2 being an R-linear sum of Di,j .
(b) Otherwise, v(x
m2
n2 x
m1
n1 z2) = v(ω2). In this case, we get
γ3 = γ̃3 − x
m2
n2 (γ2B1 − γ̃2)B2.
By Lemma 4.5, ω3 = Π3i=1x
mi
ni z3Π
3
i=1Bi + C3, with C3 an R-linear sum of
Di,j .
3. If v(ω2) > v(ω1), then v(x
m2
n2 x
m1
n1 z2) = v(ω1) and γ3 = −x
m2
n2 (γ2B1 − γ̃2)B−11 .
By Lemma 4.5, ω2 = Π2i=1x
mi
ni z3Π
2
i=1Bi + C4, with C4 an R-linear sum of Di,j .
L. Vukšić: Valuations and orderings on the real Weyl algebra 265
The general step of the evaluation is similar to the first three. Suppose that in the
previous steps, we have evaluated v(z1) = r2, . . . , v(zℓ−1) = rℓ ∈ Q. In the last step, we
have, by considering (ωi)ki=−1 for some k, begun to evaluate v(zℓ) and we are, with
ωk = (Π
k
i=1x
mi
ni )zℓ+1(Π
k
i=1Bi) + C,
where C is as in Lemma 4.5, in one of the five situations:
1. If v(ωk) < v(C), then v(zℓ) = v(ωk) −
∑k
i=1
mi
ni
and γℓ+1 = ˜γℓ+1Πki=1Bi
−1
.
In case rℓ+1 = v(zℓ) ∈ Q, our next step is to evaluate the v-value of zℓ+1 =
zℓ − rℓ+1γℓ+1 by writing
ωk+1 = (Π
k+1
i=1 x
mi
ni )zℓ(Π
k+1
i=1Bi) + C1.
2. If v(ωk) > v(C), then v(zℓ) = v(C) and γℓ+1 = CΠki=1Bi
−1
. In case rℓ+1 =
v(zℓ) ∈ Q, our next step is to evaluate the v-value of zℓ+1 = zℓ − rℓ+1γℓ+1 by
writing
ωk = (Π
k
i=1x
mi
ni )zℓ+1(Π
k
i=1Bi) + C2.
3. If v(ωk) = v(C) ̸∈ Q, then v(zℓ) = v(ωk − C) −
∑k
i=1
mi
ni
̸∈ Q since in case
v(ωk − C) > v(ωk), C ∼ ωk must hold, but since, given that C ̸= ωk and that C
is a sum of Di,j , v(ωk − C) must be in Q. This terminates our evaluation of the
sequence (zi)i≥0 associated to v’s extension to R[y; δ].
4. If v(ωk) = v(C) =
mk+1
nk+1
∈ Q and x
mk+1
nk+1 ωk = x
mk+1
nk+1 C, v((Πki=1x
mi
ni )zℓ) >
v(ωk). We continue with our evaluation by writing
ωk+1 = (Π
k+1
i=1 x
mi
ni )zℓ(Π
k+1
i=1Bi) + C3.
5. If v(ωk) = v(C) =
mk+1
nk+1
∈ Q and x
mk+1
nk+1 ωk ̸= x
mk+1
nk+1 C, then v((Πki=1x
mi
ni )zℓ) =
v(ωk) and γℓ+1 = (x
mk+1
nk+1 ωk − x
mk+1
nk+1 C)Πki=1Bi
−1
. We write
ωk+1 = (Π
k+1
i=1 x
mi
ni )zℓ+1(Π
k+1
i=1Bi) + C4.
For each 1 ≤ i ≤ 4, Ci is, as is C, an R-linear sum of Di,j for various i, j. This is assured
by Lemma 4.5.
We point out that for each ℓ, v(zℓ) is determined in a finite number of steps. In case v is
determined by a finite sequence (ωi)Ni≥−1 for some N ≥ 0, this is immediate. In the infinite
case, it follows from Lemma 3.2 that limk→∞ v(ωk) = 0, so, given that by Lemma 4.4,
v(C) is a sum of v(ωi), v(ωk) < v(C) must hold for some k. And since, in the step where
v(zℓ) is determined, we get v(Πki=1x
mi
ni zℓ) = v(zℓ) −
∑k
i=1 v(ωi−1) ≤ v(ωk) for each
ℓ ≥ 0, we get v(zℓ) ≤
∑k
i=0 v(ωi) < 1 by Lemma 3.2. Since for each ℓ ≥ 0, zℓ+1 =
zℓ − x−rℓ+1γℓ+1, we did indeed find a unique sequence (zi)i≥0 that uniquely determines a
valuation v on R[y; δ]. This valuation is v’s extension from A1(R) to R[y; δ].
The construction introduced in the proof of Theorem 4.6 can be reversed. Given a val-
uation v on R[y; δ], we could use the reverse construction to find the sequence (ωi)i≥−1 ⊆
A1(R) associated to v’s restriction to A1(R).
266 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
5 Valuations on R̃[y; δ]
The ring R̃[y; δ] is an extension of R[y, δ] where R̃ is defined as R((xQ)), the generalized
power ring of sums
∑
q∈Q αqx
−q with well-ordered support. We first show that every
valuation on R[y; δ] can be easily extended to R̃[y; δ].
Lemma 5.1. Every valuation on R[y; δ] with residue field R can be extended to a valuation
on R̃[y; δ] with the same residue field.
Proof. Suppose first v is defined on R[y; δ] by a finite sequence (zi)ki=0 with v(zk) ̸∈ Q.
Then, as we can write every f ∈ R̃[y; δ] as
∑n
i=0 pi(x)z
i
k with pi(x) ∈ R̃, we define
v(f) = min1≤i≤n{v(pi(x)) + iv(zk)}. This gives us a well-defined valuation on R̃[y; δ]
which clearly extends the one we defined on R[y; δ] in the previous section.
Now suppose v is defined by an infinite sequence (zi)i≥0 with r = limi→∞ v(zi) ≤ 1.
Define z := y−
∑∞
i=1 αix
−ri ∈ R̃[y; δ]. In the same way as before, write every f ∈ R̃[y; δ]
as
∑n
i=0 pi(x)z
i with pi(x) ∈ R̃ and define v(f) = min1≤i≤n{v(pi(x))y + i(r − µ)}
where µ is a positive infinitesimal. Thus we once more get v’s extension to R̃[y; δ].
In case r = 1, this is the only possible extension up to isomorphism of the value group,
for v(z) must be 1 − µ. This is because on the one hand, since v(z) > v(zk) = rk+1
for all k ≥ 0, v(z) is greater than any rational number q < 1. On the other hand, since
v[y− z, x] = v[y, x] = 0 and the value group is commutative, v(z) ≤ 1. Thus if v(z) ∈ R,
v(z) = 1. But if we restricted v to the quotient ring of the R-algebra, generated by x and
z, we would get a valuation on a division ring, isomorphic to D1(R), with a rational value
group, residue field R and v[z, x] = v(zx), which contradicts Corollary 2.4.
Proposition 5.2. Let v be a valuation on R̃[y; δ] with residue field R. Then the value group
is not of rational rank one.
Proof. The only case in the proof of Lemma 5.1 where it does not immediately follow
that the value group is not Q is when v’s restriction to R[y; δ] is constructed by an infinite
sequence (zi)i≥0 with r := limi→∞ v(y − zi) < 1. In this case, we can set v(z) = r ∈ R
and if r ∈ Q, define y(1) := z and restart the construction of v. We may get another
infinite sequence (z(1)i )i≥0 with r
(1) := limi→∞ v(z
(1)
i ) < 1. If r
(1) ∈ Q, we start
over with y(2) := z(1). Though we may have to repeat the process infinitely many times,
the set {z(j)k }j≥1,k≥0 is countable and A := {v(z
(j)
k )}j≥1,k≥0 is a well-ordered set of
rational numbers smaller than one. At one point, v(z) will have to be irrational for some
z =
∑
q∈A αqx
−q since we would otherwise get z ∈ R̃ such that v(z) = 1 which, as we
have shown, contradicts the fact that the value group is rational.
Recall that a pseudo-Cauchy sequence in a division ring D with a valuation v is a
sequence (aλ)λ∈Λ ⊆ D, where Λ is an ordinal such that there exists λ ∈ Λ for which
v(aσ − aρ) < v(aρ − aτ ) for all σ, ρ, τ ∈ Λ with λ ≤ σ < ρ < τ . Let D′ be an extension
of D and v′ and extension of v to D′. Then a ∈ D′ is a limit of the pseudo-Cauchy
sequence (aλ)λ∈Λ if v′(a − aσ) = v′(aσ+1 − aσ) for all σ ∈ Λ, λ ≤ σ. Recall also the
definition of a immediate extension of a valuation.
L. Vukšić: Valuations and orderings on the real Weyl algebra 267
Definition 5.3. Let v be a valuation on a division ring D, the division ring D′ an extension
of D and v′ a valuation on D′ that extends v. Let Γ and D and Γ′ and D′ be the value
groups and residue division rnigs associated to v and v′. Then we say v′ is an immediate
extension of v if [Γ′ : Γ] = 1 and [D′ : D].
As a byproduct of our investigations, we show that not every extension of a valued
division ring by limits of pseudo-Cauchy sequences is immediate. This differs from the
commutative case since, as Kaplansky proved in [8], every extension of a valued field by
limits of pseudo-Cauchy sequences is immediate.
Corollary 5.4. There exist division rings D ⊆ D′ and a valuation v on D which extends
to a valuation v′ on D′ such that D′ is an extension of D by limits of pseudo-Cauchy
sequences in D whereas v′ is not an immediate extension of v.
Proof. Let v be a valuation on R[y; δ] with residue field R and value group Q as described
in Section 4. Then v(x) = −1 and R̃[y; δ] is an extension of R[y; δ] by limits of pseudo-
Cauchy sequences. This holds because every
∑
i∈Q aix
−i ∈ R̃ is a limit of the pseudo-
Cauchy sequence (
∑k
i=1 aix
−qi)k≥1 in R[y; δ].
As we have shown in this section, v can be uniquely extended to R̃[y; δ]. By Proposi-
tion 5.2, this extension is not immediate. Let D be the quotient division ring of R[y, δ], to
which v uniquely extends, and D′ the quotient division ring of R̃[y; δ], to which v′ uniquely
extends, since both rings are Ore domains. D′ is not an immediate extension of the ring
D with valuation v, even though D′ is an extension of D by limits of pseudo-Cauchy
sequences.
6 Compatibility with orderings on A1(R) and R[y; δ]
In Section 3, we mentioned that every strongly abelian valuation on a division ring with
an ordered residue field is compatible with an ordering on the valued division ring. In
this section, we will use a noncommutative version of the Baer-Krull theorem to determine
all orderings on A1(R) compatible with one of the valuations v we have described in the
previous sections. We will then show which of these orderings on A1(R) can be extended
to an ordering on R[y; δ] compatible with a v’s extension to R[y; δ].
Recall that an order P on a division ring D is compatible with a valuation v on D if for
every a, b ∈ D∗ such that v(a) = v(b) < v(a− b), ab ∈ P holds.
Let v be a strongly abelian valuation on a division ring D with a formally real residue
field D. Let Γ be its value group. Let s : Γ → D∗ be a semisection of v, i.e., a map for
which
1. s(0) = 1,
2. v(s(g)) = g for all g ∈ Γ,
3. s(g1 + g2) = s(g1)s(g2)u2 for some u ∈ D∗ for all g1, g2 ∈ Γ.
Let χ : Γ/2Γ → {−1, 1} be a group homomorphism called a character and let P be an
ordering of D. Then, as it was shown in [24],
Pχ = {a ∈ D | a · s(v(a))−1χ(v(a) + 2Γ) ∈ P}
268 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
is an order of D compatible with v. Moreover, if X denotes all orders of the residue field,
Xv denotes all v-compatible orders on D and (Γ/2Γ)∗ denotes the set of all characters on
Γ/2Γ, then by Proposition 3 of [24], the map
f : X × (Γ/2Γ)∗ → Xv
f((P , χ)) = Pχ
is a bijection. The choice of a semisection s on Γ does not matter. Using f , we will now
describe all orders on A1(R) that are compatible with a valuation described in Section 3.
Suppose v is a valuation on A1(R) associated to an infinite sequence (ωi)i≥−1 with R
as a residue field. There is only one possible order of R, so the orders of A1(R) compatible
with v will only depend on the characters χ : Γ/2Γ → {−1, 1}. Then there are three
different options for the value group Γ ⊆ Q× Z.
1. Γ is a 2-divisible subgroup of Q,
2. Γ is a non-2-divisible subgroup of Q,
3. Γ is a direct sum of a non-2-divisible subgroup of Q with Z.
Since in each case the value group Γ is generated by {v(ωi)}i≥−1, the characters and thus
the v-compatible orders will be determined by the signs of the ωi.
In the first two cases, v is determined by an infinite sequence (ωi)i≥−1 with v(ωi) =
mi+1
ni+1
∈ Q for each i ≥ −1. In the third case, v is determined by a finite sequence (ωi)Ni=−1
with v(ωi) =
mi+1
ni+1
∈ Q for each −1 ≤ i ≤ N − 1 and v(ωN ) ̸∈ Q.
If the value group is a 2-divisible subgroup of Q, then for each ωi, there is a j ̸= i such
that Ki,j ∈ Z is odd while Kj,i ∈ Z is even. Since ω
Ki,j
i−1 ω
Kj,i
j−1 = αi,j , it follows that
ωi−1 > 0 if and only if αi,j > 0.
Conversely, if the value group is a non-2-divisible subgroup of Q, we choose one i
such that ni is divisible by the greatest power of two that divides nj for any j ≥ 0. After
choosing either ωi < 0 or ωi > 0, the order on A1(R) is defined.
In the last case, where the value group is a direct sum of a non-2-divisible subgroup of
Q and Z, the order is determined by choosing either ωi < 0 or ωi > 0 and, independently,
either ωN > 0 or ωN < 0 where i is as in the second case and v(ωN ) ̸∈ Q. All four
combinations define an ordering on A1(R).
We have thus proven the following proposition.
Proposition 6.1. Suppose v is a valuation on A1(R) with residue field R and value group
Γ. Then:
1. If Γ is a 2-divisible subgroup of Q, there is a unique v-compatible ordering on
A1(R).
2. If Γ is a non-2-divisible subgroup of Q, there are two v-compatible orderings on
A1(R).
3. If Γ is a direct sum of a non-2-divisible subgroup of Q with Z, there are four possible
v-compatible orderings on A1(R).
L. Vukšić: Valuations and orderings on the real Weyl algebra 269
6.1 Extensions of orderings on A1(R) to orders on R[y; δ]
In this section, we show which orders on A1(R) are extendable to an order on R[y; δ],
thereby answering the question posed by Marshall and Zhang in [15].
Every order on A1(R) is compatible with a unique finest valuation v on the same ring
with residue field R, as proved in [15]. Suppose v is a valuation on A1(R) associated to
an infinite sequence (ωi)i≥−1 with v(ωi−1) = mini and ωi = x
miωnii−1 − βi. In Section 4,
we showed that provided both conditions of Theorem 4.6 are fulfilled, v can be uniquely
extended to a valuation v′ on R[y; δ] with residue field R if the value group is 2-divisible
and that it has two extensions to R[y; δ] with the same residue field if the value group is
non-2-divisible. Here we show all v-compatible orders on A1(R) we have described in
the first part of this section can be extended to a v′-compatible order on R[y; δ] for some
extension v′ of v from A1(R) to R[y; δ].
Theorem 6.2. Let P be an ordering on A1(R) and v be the unique finest valuation on
A1(R) compatible with P . Then:
1. The order P can be extended to an ordering on R[y; δ] if and only if v can be extended
to a valuation on R[y; δ] with residue field R.
2. If the v-value group Γ is a 2-divisible subgroup of Q, then the extension of P is
unique. If on the other hand, Γ is a subgroup of Γ1 × Z, where Γ1 is a non-2-
divisible subgroup of Q, i.e., when Γ is not a 2-divisible subgroup of Q, there are
two extensions of P to R[y; δ]. Each of the two extensions of v to a valuation v′ on
R[y; δ] with residue field R uniquely determines one of the two of P ’s extensions to
R[y; δ].
Proof. The first statement of the theorem follows from the fact that every ordering on
R[y; δ] is compatible with a valuation on the same ring with residue field R.
To prove the second statement, suppose that v is the unique P -compatible valuation on
A1(R) with residue field R that extends to a valuation on R[y; δ] with the same residue
field.
If the value group Γ of v on A1(R) is either a 2-divisible or non-2-divisible subgroup
of Q, then the value group Γ′ of v’s extension to R[y; δ] is Q. In this case, there is exactly
one v′-compatible order of R[y; δ] for each of v’s extensions v′ to R[y; δ].
Suppose Γ is a 2-divisible subgroup of Q. Then there is a unique extension v′ of v to
R[y; δ]. It follows that in case Γ is a 2-divisible subgroup of Q, the only v-compatible order
on A1(R) extends to an order of R[y; δ] that is compatible with v′.
If, on the other hand, Γ is a non-2-divisible subgroup of Q, there are two extensions v′
of v to R[y; δ]. We will now show that for each of the v-compatible orderings on A1(R),
there is a unique extension v′ of v to R[y; δ] such that the ordering on A1(R) can be
extended to the unique v′-compatible ordering on R[y; δ].
In this case, a v-compatible ordering on A1(R) is, as we have shown in the beginning
of this section, uniquely determined by the sign of ωi−1 where i ≥ 1 is such that ni is
divisible by the greatest power of two that divides nj for any j ≥ 1. Furthermore, v′, the
extension of v to R[y; δ], is uniquely determined by choosing the sign of γ̃i for this i.
We first choose an extension v′ of v to R[y; δ]. We observe that x
m
n > 0 must hold for
every mn ∈ Q since all rational powers of x are in R[y; δ]. Since x
mi
ni ωi−1 = γ̃i for each
i ≥ 1, γ̃iωi−1 > 0 must hold for all i ≥ 0 for the order to be extendable to a v-compatible
270 Ars Math. Contemp. 24 (2024) #P2.04 / 231–271
order on R[y; δ]. This holds for exactly one of the two v-compatible orders on A1(R). It is
clear from the construction that for each ordering on A1(R), there is exactly one extension
v′ of v to R[y; δ] such that this ordering is extendable to the unique v′-compatible ordering
on R[y; δ].
In case Γ is a subgroup of Q× Z of rational rank two, Γ′ = Q× Z holds. In this case,
there are two v′-compatible orderings on R[y; δ] for every extension v′ of v to R[y; δ]. We
will now show that for each of the four v-compatible orders on A1(R), there is a unique
extension v′ of v to R[y; δ] and a unique v′-compatible ordering P ′ on R[y; δ] such that P ′
is an extension of P . The ordering P compatible to a valuation v on A1(R) is determined
by the signs of ωi−1 and ωN where i ≥ 1 is such that ni is divisible by the greatest power of
two that divides nj for any j ≥ 1, and v(ωN ) ̸∈ Q. The extension v′ of v to R[y; δ] and the
v′-compatible ordering on R[y; δ] that extends P are the valuation v′ for which ωi−1γ̃i > 0
and the v′-compatible ordering that agrees with the signs of ωi−1 and ωN in P .
We have thus proved the second statement of the theorem.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.05 / 273–292
https://doi.org/10.26493/1855-3974.2764.fe5
(Also available at http://amc-journal.eu)
Regular dessins with moduli fields of the form
Q(ζp, p
√
q)*
Nicolas Daire
Département de Mathématiques et Applications, École Normale Supérieure,
45 rue d’Ulm, 75005 Paris, France
Fumiharu Kato , Yoshiaki Uchino
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro,
Tokyo 152-8551, Japan
Received 10 December 2021, accepted 29 March 2023, published online 27 September 2023
Abstract
Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins
with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli
fields are nonabelian Galois extensions of the form Q(ζp, p
√
q), where p is an odd prime
and ζp is a pth root of unity and q ∈ Q is not a pth power, and we then show that their
regular closures have the same moduli fields. Finally, in the special case p = q = 3 we
give another example of a regular dessin of degree 219 ·34 and genus 14155777 with moduli
field Q(ζ3, 3
√
3).
Keywords: Dessins d’enfants, coverings.
Math. Subj. Class. (2020): 14H57, 14H30
1 Introduction
Grothendieck first coined the term Dessin d’enfant in Esquisse d’un Programme [4] to
denote a connected bicolored graph embedded on a compact connected oriented topolog-
ical surface. The study was motivated by the one to one correspondance between dessins
d’enfant, the combinatorial data of the associated cartographical group, and the geomet-
ric concept of coverings of P1 by compact Riemann surfaces ramified at most over three
*The authors are grateful to Professor Jürgen Wolfart for valuable comments.
E-mail addresses: nicolas.daire@ens.psl.eu (Nicolas Daire), bungen@math.titech.ac.jp (Fumiharu Kato),
uchino.y.ab@m.titech.ac.jp (Yoshiaki Uchino)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
274 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
points. Moreover, by Belyi’s theorem any such covering is given the structure of an alge-
braic curve defined over a number field, therefore we obtain a natural action of the absolute
Galois group Gal(Q̄/Q) on the set of isomorphism classes of dessins. A lot of the interest
for dessins stems from the fact that this action is faithful, providing a way to study the
absolute Galois group through its action on the set of dessins. A particularly interesting
family of dessins is that of regular dessins, characterized by the fact that their automor-
phism groups act transitively on their sets of edges, and the Galois action was proved to
remain faithful when restricted to the subset of isomorphism classes of regular dessins [3].
To any dessin we associate a number field called its moduli field, which is defined as the
subfield of Q̄ fixed by the subgroup of Gal(Q̄/Q) that fixes the dessin up to isomorphism.
Conder, Jones, Streit and Wolfart noted in [1] that the moduli fields of all the examples
of regular dessins known at the time were abelian Galois extensions of Q. Herradón con-
structed in [6] an explicit equation for a regular dessin whose moduli field Q( 3
√
2) is not
a Galois extension of Q, and Hidalgo later generalized his construction in [7] to produce
regular dessins whose moduli fields are of the form Q( p
√
2) where p is an odd prime num-
ber. However there is as of yet no known example of regular dessin whose moduli field
is a nonabelian Galois extension of Q. This is the starting point of this paper, in which
we will exhibit examples of regular dessins with moduli fields that are nonabelian Galois
extensions of Q.
In the present paper, we begin by recalling the main definitions and results on dessins
d’enfant. We will then expose constructions of regular dessins whose moduli fields are
nonabelian Galois extensions of Q. We first exhibit dessins whose moduli fields are of
the form Q(ζ3, 3
√
q), where ζ3 is a primitive third root of unity and q ∈ Q is not a third
power, and show that the regular closures of these dessins possess the same moduli fields.
We then generalize this construction to show that there exist regular dessins with moduli
fields Q(ζp, p
√
q), where ζp is a primitive pth root of unity and q ∈ Q>0 is not a pth power.
Finally, we give an example of a regular dessin of degree 219 ·34 and genus 14155777 with
moduli field Q(ζ3, 3
√
3).
Notations
• SE : the group of self-bijections of the set E, similarly Sn is the group of permu-
tations of a set of n elements (we favor a right action, hence we write the product
στ := τ ◦ σ)
• Gal(E/F ): the Galois group of F -automorphisms of E
• ζk: the kth primitive root of unity exp( 2iπk )
• F2: the free group of rank 2 with generators (ξ, η)
• Crit: the set of critical values of a function
2 Preliminaries on dessins d’enfant
We refer the reader to existing expositions of the theory such as [5, 8, 9] and [2] for proofs
of the presented facts and further details.
A dessin d’enfant is a connected bipartite graph embedded on a compact connected
orientable topological surface, such that the complement of the graph is a disjoint union of
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 275
2-cells. Two such dessins are equivalent if there exists an orientation preserving homeomor-
phism between the underlying surfaces that induces an isomorphism between the embedded
bipartite graphs.
A dessin is determined up to isomorphism by a pair (C, β) where C is a smooth alge-
braic curve and β : C → P1 is a meromorphic mapping ramified at most over {0, 1,∞},
and by Belyi’s theorem we can further ask for C and β to both be defined over a number
field. We call (C, β) a Belyi pair and β a Belyi function. The corresponding graph embed-
ding on the underlying surface is recovered by pulling back the segment [0, 1] along β, we
define black and white vertices as the preimages of 0 and 1 respectively, and the edges as
the preimages of ]0, 1[.
By covering theory a dessin is also determined up to isomorphism by the monodromy
action of the fundamental group of the complex projective line π1(P1) on the fiber over the
point 12 which is identified to the set of edges of the dessin. The fundamental group π1(P
1)
is isomorphic to the free group of rank two F2 = ⟨ξ, η⟩ with generators ξ and η which are
two loops with base point 12 and circling counter-clockwise around 0 and 1 respectively.
The monodromy action of the generators ξ and η then corresponds to the product of the
counter-clockwise cyclic permutation of the edges around black and white vertices respec-
tively. We call monodromy map M : F2 → SE the map that associates to each element of
F2 the corresponding permutation of the set of edges, and we call cartographic group the
image of the monodromy map, which is a transitive subgroup of the group of permutations
of the set of edges.
When the automorphism group of a dessin D acts transitively on the set of edges, we say
that D is a regular dessin. When that is the case the cartographic group G acts transitively
and freely on the set of edges, the monodromy action is thus given by the canonical action of
G on itself. There is a natural bijection between regular dessins and finite groups generated
by two distinguished elements ξ and η up to isomorphism. Two regular dessins determined
by G1 = ⟨ξ1, η1⟩ and G2 = ⟨ξ2, η2⟩ respectively are isomorphic if and only if there exists
an isomorphism between G1 and G2 that preserves the distinguished generators. Given
a dessin D, there exists a unique regular dessin D̃ with a morphism ϕ : D̃ → D such
that any morphism from a regular dessin to D factors through ϕ. We call D̃ the regular
closure of D. Moreover, there exists an isomorphism Cart(D̃) ∼= Cart(D) that preserves
the distinguished generators. There exists a natural action of the absolute Galois group
Gal(Q̄/Q) on the set of isomorphism classes of dessins, we denote by Dσ the action of an
automorphism σ on a dessin D, and this Galois action commutes with regular closure, i.e.
we have (D̃)σ ∼= (̃Dσ).
Given a dessin D, we say that a number field k is a field of definition of D if D is
isomorphic to a dessin defined over k. However there does not necessarily exist a smallest
field of definition. We thus define the moduli field of a dessin D as the subfield of Q̄ fixed
by the subgroup of Gal(Q̄/Q) constituted of the elements fixing D up to isomorphism. The
moduli field of a dessin is contained in all fields of definition but is not necessarily itself a
field of definition, however it is the case in particular for regular dessins.
3 Constructions of regular dessins with nonabelian moduli fields
We are now ready to give examples of regular dessins whose moduli fields are nonabelian
Galois extensions of Q. To do so, we will first exhibit dessins with such moduli fields, and
then prove that their regular closures admit the same moduli fields.
276 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
Before proceeding with the examples, let us first present a classic family of Belyi poly-
nomials that we will use in the following constructions. For positive integers m,n ∈ N we
define the polynomial
Bm,n :=
(m+ n)m+n
mmnn
Xm(1−X)n ∈ Q[X].
By computing the derivative B′m,n =
(m+n)m+n
mmnn X
m−1(1 −X)n−1(m − (m + n)X) we
verify that Bm,n : P1 → P1 is a Belyi function that ramifies only at 0, 1, ∞ and mm+n with
ramification indices m, n, m + n and 2 respectively, and Bm,n(0) = 0, Bm,n(1) = 0,
Bm,n(∞) = ∞ and Bm,n( mm+n ) = 1 (see Figure 1).
0 1
m
m+n
1
2
3
m − 1
m
1
2
3
n − 1
n
Figure 1: Dessin corresponding to the Belyi pair (P1, Bm,n).
3.1 Regular dessins with moduli fields of the form Q(ζ3, 3
√
q)
Let q ∈ Q>0 be a positive rational number that is not a third power.
Let m,n ∈ N be coprime positive integers such that 2727+q2 = mm+n , and let
C : y2 = x(x− (1− ζ3))(x− 3
√
q),
β : C → P1, (x, y) 7→ 1
27mq2n
(x6 + 27)m(q2 − x6)n.
The function β is given by the composition β = β1 ◦ β0 ◦ π of the following maps.
1. π : C → P1 is the projection on the coordinate x, which is ramified over {0, 1 −
ζ3, 3
√
q,∞}.
2. β0 := X6 ∈ Q[X], Crit(β0) = {0} so β1 ◦ π ramifies over {0, (1 − ζ3)6 =
−27, q2,∞}.
3. β1 := Bm,n(X+27q2+27 ), so β = β1 ◦ β0 ◦ π ramifies over {0, 1,∞}.
The pair (C, β) is thus a Belyi pair, and we call D the corresponding dessin. The dessin
D is defined over Q(ζ3, 3√q), so its moduli field is a subfield of Q(ζ3, 3√q). By taking
the regular closure we then obtain the inclusion of moduli fields M(D̃) ⊆ M(D) ⊆
Q(ζ3, 3
√
q), and moreover D̃ is regular so it is defined over M(D̃). We shall prove that
M(D̃) is in fact exactly Q(ζ3, 3√q), which is a nonabelian Galois extension of Q with
Galois group
Gal(Q(ζ3, 3
√
q)/Q) ∼= S3.
To that end we must show that an automorphism σ ∈ Gal(Q̄/Q) fixes D̃ if and only if it
fixes ζ3 and 3
√
q, or equivalently that Gal(Q(ζ3, 3
√
q)/Q) acts freely on the orbit of D̃.
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 277
Let σ ∈ Gal(Q̄/Q), the Galois conjugate Dσ is given by the Belyi pair (Cσ, βσ),
where
Cσ : y2 = x(x− (1− σ(ζ3)))(x− σ( 3
√
q)),
and βσ has the same expression as β because all of its coefficients are rational. The orbit
of the pair (ζ3, 3
√
q) by Gal(Q̄/Q) is {ζi3, ζj3 3
√
q}1≤i≤2,0≤j≤2. Elliptic curves given by
equations of the form y2 = (x− a)(x− b)(x− c) are isomorphic if and only if the cross-
ratios of the tuples (a, b, c,∞) coincide. We verify that the cross-ratios are all distinct, so
the orbit of D is given by the six dessins Dσ for σ ∈ Gal(Q(ζ3, 3√q)/Q). As a consequence
M(D) = Q(ζ3, 3√q). To prove that the regular closures D̃σ constituting the orbit of D̃ are
also non isomorphic, we must first draw the dessins Dσ to compute their cartographic
groups.
Let us first draw the dessin D0 corresponding to the Belyi pair (P1, β1 ◦ β0) (see Fig-
ure 3). The dessin D0 is defined over Q, so the dessins Dσ in the orbit are then obtained
by lifting D0 to the curves Cσ . To simplify the graphical representations of the dessins, we
will use the notation in Figure 2 for consectutive edges incident to a vertex.
eξ
eξ
2
eξ
n−2
eξ
n−1
eeξ
n
=
n
eeξ
n
Figure 2: Notation for consecutive edges.
(P1, id)0 1
(P1, β1)
−27 0 q2
m n
(P1, β1 ◦ β0)
0
n
m
n
m
n
m
n
m
n
m
n
m
3
√
q
ζ3 3
√
q
ζ23 3
√
q
1− ζ23
1− ζ3
Figure 3: Construction of D0.
The dessins D1, . . . ,D6 conjugate to D are embedded on a torus, so in the representa-
tions in Figure 4 we will identify the outermost edges on opposite sides.
278 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
?
∞
0 0
00
3
√
q 3
√
q
1− ζ3
1− ζ3
n
4
m
3
n
16
m
15
n
6
m
5
n
18
m
17
n
8
m
7
n
20
m
19
n
10
m
9
n
22
m
21
n
12
m
11
n
24
m
23
n n
m
m
1 13
14
2
131
2
14
(a) D1 := D
?
∞
0 0
00
ζ3 3
√
q ζ3 3
√
q
1− ζ23
1− ζ23
n
2
m
3
n 14
m
15
n
6
m
5
n
18
m
17
n
8
m
7
n
20
m
19
n
10
m
9
n
22
m
21
n
12
m
11
n
24
m
23
n n
m
m
1 13
16
4
131
4
16
(b) D2 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , ζ3 3
√
q)
?
∞
0 0
00
ζ3 3
√
q ζ3 3
√
q
1− ζ3
1− ζ3
n2
m
3
n 14
m 1
5
n
4
m
5
n
16
m
17
n
8
m
7
n
20
m
19
n
10
m
9
n
22
m
21
n
12
m
11
n
24
m
23
n n
m
m
1 13
18
6
131
6
18
(c) D3 := Dσ, σ : (ζ3, 3√q) 7→ (ζ3, ζ3 3√q)
?
∞
0 0
00
ζ23 3
√
q ζ23 3
√
q
1− ζ23
1− ζ23
n2
m
3
n 14
m 15
n
4
m
5
n
16
m
17
n
6
m
7
n
18
m
19
n
10
m
9
n
22
m
21
n
12
m
11
n
24
m
23
n n
m
m
1 13
20
8
131
8
20
(d) D4 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , ζ23 3
√
q)
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 279
?
∞
0 0
00
ζ23 3
√
q ζ23 3
√
q
1− ζ3
1− ζ3
n2
m3
n 14
m 15
n
4
m
5
n 1
6
m
17
n
6
m
7
n
18
m
19
n
8
m
9
n
20
m
21
n
12
m
11
n
24
m
23
n n
m
m
1 13
22
10
131
10
22
(e) D5 := Dσ, σ : (ζ3, 3√q) 7→ (ζ3, ζ23 3
√
q)
?
∞
0 0
00
3
√
q 3
√
q
1− ζ23
1− ζ23
n2
m3
n 14
m 15
n
4
m
5
n 16
m 1
7
n
6
m
7
n
18
m
19
n
8
m
9
n
20
m
21
n
10
m
11
n
22
m
23
n n
m
m
1 13
24
12
131
12
24
(f) D6 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , 3
√
q)
Figure 4: Dessins D1, . . . ,D6 in the Galois orbit of D.
We will now establish that D̃1 is not isomorphic to D̃2, . . . , D̃6. To that end it suffices
to show that there is no isomorphism between the cartographic groups fixing the canonical
generators. We shall therefore exhibit an element ω ∈ F2 = ⟨ξ, η⟩ such that Mk(ω)
commutes with Mk(η2) only when k = 1, where Mk is the monodromy map of Dk.
We have defined m and n to be positive coprime integers such that 2727+q2 =
m
m+n , so
we cannot have m = n = 1. We will treat the case where m ̸= 1 does not divide n, the
other case being treated similarly. Let
ω := ξnη−1ξm−nηξn.
We shall show that Mk(ω) commutes with Mk(η2) only when k = 1.
Let Ek := {1, 2, . . . , 24} be the set of edges of Dk incident to 0. The action of η fixes
the set Ek on which it induces the cyclic permutation (1, 2, . . . , 24), and every white vertex
except 0 has degree one so the action of η is trivial on the complement of Ek.
We can write Ek = Eoddk ⊔ Eevenk as the disjoint union of the sets of respectively
odd and even numbered edges incident to 0, such that η sends one to the other. The black
vertices of Eoddk are of degree m except for the two black vertices of the edges 1 and 13 that
are of degree 2m. Therefore if m does not divide some integer l then ξl sends every edge of
Eoddk to the complement of Ek, and otherwise the action of ξ
m on Eoddk corresponds to the
sole transposition (1, 13). Similarly if n does not divide l then ξl sends every edge of Eevenk
to the complement of Ek, and the action of ξn on Eevenk is the transposition (2k, 2k + 12).
In particular, by hypothesis n is not a multiple of m, so m − n is not a multiple of m
either, hence both ξn and ξm−n send the edges of Eoddk to the complement of Ek. However
η acts trivially on the latter, so ξnη−1ξm−n and ξm−nηξn both fix the set Eoddk on which
they induce the same action as ξm, i.e. the transposition (1, 13). Therefore the action
of ω = ξnη−1ξm−nηξn is the same as that of ξmηξn on Eoddk and the same as that of
ξnη−1ξm on Eevenk . See Figure 5.
The action of ω fixes the set Ek on which it induces the permutation
Mk(ω)|Ek = (1, 13)(2k, 2k + 12) · (1, 2)(3, 4) · · · (23, 24) · (1, 13)(2k, 2k + 12)
280 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
0
m
ξn
η−1
ξm−n
n ξn
η
e
eω
0
m
ξm−n
η
ξn
n ξn
η−1
eω
e
Figure 5: Action of ω on Eoddk \ {1, 2k − 1} and on Eevenk \ {2, 2k}.
Therefore for k = 1,
M1(ω)|E1 = (1, 13)(2, 14) · (1, 2)(3, 4) · · · (23, 24) · (1, 13)(2, 14)
= (1, 2)(3, 4) · · · (23, 24)
so ω and η2 commute on E1. Moreover η acts trivially on the complement of E1 so
M1(ω)|D1\E1 and M1(η2)|D1\E1 automatically commute. Finally, we obtain that M1(ω)
and M1(η2) commute.
For k = 2, we observe that 4ωη
2
= 15η
2
= 17 but 4η
2ω = 6ω = 5. Similarly, for
3 ≤ k ≤ 6, we observe that 1ωη2 = 14η2 = 16 but 1η2ω = 3ω = 4. We have thus shown
that Mk(ω) and Mk(η2) commute only for k = 1.
This concludes the proof that D̃ is a regular dessin with moduli field Q(ζ3, 3√q).
3.2 Regular dessins with moduli fields of the form Q(ζp, p
√
q)
Let p be an odd prime, and q ∈ Q>0 a positive rational number that is not a pth power. In
this example we will need an additional parameter γ ∈ Q \ {0}. Let
C : y2 = x(x− (1− ζp))(x− γ p
√
q).
We construct the Belyi function β : C → P1 as the composition β = β2 ◦ β1 ◦ β0 ◦ π
of the following maps.
1. π : C → P1 is the projection on the coordinate x, which ramifies over {0, 1 −
ζp, γ p
√
q,∞}.
2. β0 := X2p ∈ Q[X], and Crit(β0) = {0,∞} so β0 ◦ π ramifies over {0, (1 −
ζp)
2p, γ2pq2,∞}.
3. β1 ∈ Q[X] is chosen independently of γ such that Crit(β1) ∪ {β1((1 − ζp)2p)} =
{0, 1,∞}, β1((1−ζp)2p) = 0 < β1(0) < 1 and β′1(0) > 0. The existence of β1 veri-
fying those conditions is assured by Proposition 3.2 below. Under those assumptions
β1 ◦ β0 ◦ π ramifies over {0, 1, β1(0), β1(γ2pq2),∞}.
4. γ ∈ Q>0 is then chosen small enough so that β′1 > 0 on [0, γ2pq2]. This guarantees
us that we have 0 < β1(0) < β1(γ2pq2) < 1.
5. β2 := Br,s ◦ Bm,n, where (m,n) and (r, s) are pairs of coprime positive integers
such that β1(γ2pq2) = mm+n and Bm,n(β1(0)) =
r
r+s . Finally, β = β2 ◦ β1 ◦ β0 ◦ π
ramifies over {0, 1,∞}.
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 281
The pair (C, β) is thus a Belyi pair, and we call D the corresponding dessin. With
the same arguments as before, the moduli field of D is Q(ζp, p√q), which is a nonabelian
Galois extension of Q with Galois group
Gal(Q(ζp, p
√
q)/Q) ∼= Z/pZ ⋊ (Z/pZ)×
generated by σ : ζip p
√
q 7→ ζi+1p p
√
q and τ : ζip p
√
q 7→ ζgip p
√
q where g generates (Z/pZ)×.
We shall show that there exists γ ∈ Q \ {0} such that the regular closure of the dessin D
thus obtained also has moduli field Q(ζp, p
√
q).
Remark 3.1. In the previous subsection we treated the case p = 3. In that specific case
we gave a simpler expression for β, mainly due to the fact that β0 ◦ π already had all of its
critical values in Q∪{∞}. However in the general case we must use the intermediate map
β1 as well as the parameter γ to conclude the proof.
Let us first prove the existence of β1.
Proposition 3.2. Let E ⊂ Q̄ ∩ R \ {0} be a finite set. Then there exists P ∈ Q[X] such
that P (E) ⊆ {0}, Crit(P ) ⊆ {0, 1}, 0 < P (0) < 1 and P ′(0) > 0.
Remark 3.3. In the context of this proposition we only deal with polynomials so for P ∈
Q[X] we define Crit(P ) := {P (z)| z ∈ C, P ′(z) = 0}, which does not include the point
at infinity to simplify notations.
Proof. To show this we will proceed similarly as in the proof of the only if part of Belyi’s
theorem, by applying additional transformations to ensure that 0 < P (0) < 1. Let us first
prove that we can reduce to the case where E is a subset of rational numbers.
Lemma 3.4. Let E ⊂ Q̄ ∩ R \ {0} be a finite set fixed by Gal(Q̄/Q). Then there exists
P ∈ Q[X] such that P (0) = 0 and Crit(P ) ∪ P (E) ⊂ Q \ {0}.
Proof. Let {a1, . . . , am} = E∩Q and {b1, . . . , bn} = E\Q. We construct P by induction
on the number n of non rational elements of E.
For α ∈ Q, define Fα, Gα ∈ Q[X] by
Fα :=
n∏
j=1
(X − (bj − α)2) and Gα := Fα((X − α)2) =
n∏
j=1
(X − bj)(X + bj − 2α).
Let us first assume that there exists α ∈ Q such that Gα(0) ̸∈ Crit(Gα) ∪ Gα(E).
Define P1(X) := Gα(X)−Gα(0) ∈ Q[X], then P1(0) = 0 ̸∈ E′ := Crit(P1)∪P1(E) ⊂
Q̄ ∩ R \ {0}. Note that E′ is stable under the action of Gal(Q̄/Q), and |E′ \ Q| =
|Crit(Fα) ∪ Fα(0) \ Q| = |Crit(Fα) \ Q| < degFα = n. By induction, there exists
P2 ∈ Q[X] such that P2(0) = 0 and Crit(P2)∪P2(E′) ⊂ Q\{0}. Now P := P2 ◦P1 has
the desired properties, since P (0) = 0 and Crit(P )∪P (E) = Crit(P2)∪P2(Crit(P1))∪
P2(P1(E)) = Crit(P2) ∪ P2(E′) ⊂ Q \ {0}.
Let us now prove that there exists α ∈ Q such that Gα(0) ̸∈ Crit(Gα) ∪ Gα(E).
Let us first treat the case where 0 < b1 < b2, . . . , bn. When α approaches 12 , Gα(0) =∏n
j=1 −bj(bj−2α) approaches 0 but the critical values of Gα do not. Indeed, Crit(Gα) =
CritFα∪Fα(Crit((X−α)2)) = Crit(Fα)∪{Fα(0)}; Fα(0) approaches F b1
2
(0) ̸= 0, and
since F b1
2
does not have multiple roots, the critical values of Fα approach the critical values
282 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
of F b1
2
which are all non zero. Therefore for α ̸= b12 in the neighborhood of b12 we have
Gα(0) ̸∈ Crit(Gα). Moreover Gα(0), Gα(a1), . . . , Gα(am) are all distinct polynomials in
the indeterminate α, so they coincide at only finitely many points. In particular for α ̸= b12
in the neighborhood of b12 we have Gα(0) ̸∈ {Gα(a1), . . . , Gα(am)}. Since α ∈ Q we
also have Gα(0) ̸= 0 = Gα(b1) = · · · = Gα(bn) hence Gα(0) ̸∈ Gα(E), proving the
existence of α as desired.
Let us now treat the general case where b1, . . . , bn are not assumed to be positive by
reducing it to the previous case. For α′ ∈ Q, define Hα′ ∈ Q[X] by
Hα′ := (X − α′)2 − α′2 ∈ Q[X].
Note that Crit(Hα′) = {−α′}. For α′ > 0 sufficiently small we have −α′2 < Hα′(0) =
0 < Hα′(a1), . . . ,Hα′(am), Hα′(b1), . . . ,Hα′(bn). Let E′′ := Crit(Hα′) ∪ Hα′(E).
The set E′′ is a finite subset of Q̄ ∩ R \ {0} fixed by Gal(Q̄/Q), and E′′ has at most
n non rational elements, which are all positive. By the above, there exists P3 ∈ Q[X]
such that Crit(P3) ∪ P3(E′′) ⊂ Q \ {0} and P3(0) = 0. Then P := P3 ◦ Hα′ has the
desired properties, since P (0) = 0 and Crit(P ) ∪ P (E) = Crit(P3) ∪ P3(Crit(Hα′)) ∪
P3(Hα′(E)) = Crit(P3) ∪ P3(E′′) ⊂ Q \ {0}.
Let us denote by P1 the polynomial obtained using this lemma, which verifies P1(0) =
0 and E′ := Crit(P1) ∪ P1(E) ⊂ Q \ {0}. We can further assume that P ′1(0) > 0 by
taking (−P1) if necessary. We now send the points E′ to {0, 1}.
Lemma 3.5. Let E ⊂ Q \ {0} a finite set. Then there exists P ∈ Q[X] such that P (E) ⊆
{0}, Crit(P ) ⊆ {0, 1}, 0 < P (0) < 1 and P ′(0) > 0.
Proof. For α ∈ Q, let Fα := (X − α)2 ∈ Q[X], and note that Crit(Fα) = {0}. There
exists α < 0 sufficiently small such that 0 < Fα(0) < Fα(a) for all a ∈ E. We take
F :=
Fα
maxa∈E Fα(a)
.
Let {a1, . . . , al} = F (E) such that 0 < F (0) < a1 < · · · < al = 1. We also add a
rational point a0 ∈ Q such that F (0) < a0 < a1.
Let m and n be the coprime positive integers such that al−1 = mm+n . We recall that
Bm,n verifies Crit(Bm,n) = {0, 1}, Bm,n(0) = Bm,n(1) = 0, Bm,n( mm+n ) = 1, and
Bm,n is strictly increasing between 0 and mm+n . Let P1 := Bm,n, then Crit(P1) = {0, 1}
and 0 < P1 ◦F (0) < P1(a0) < · · · < P1(al−1) = 1. There is one point fewer than before,
so we can iteratively construct P2, . . . , Pl in the same way, so that P := Pl ◦ · · · ◦ P1
verifies Crit(P ) ⊆ {0, 1}, P (a1) = · · · = P (al) = 0 < P (F (0)) < 1 = P (a0) and
P ′(F (0)) > 0. Therefore P ◦ F has the desired properties.
Let us denote by P2 the polynomial obtained using this lemma with the finite set E′
obtained previously. Then the polynomial P := P2 ◦ P1 verifies P (E) ⊆ {0}, Crit(P ) ⊆
{0, 1}, 0 < P (0) < 1 and P ′(0) > 0, thus concluding the proof of Proposition 3.2.
We can now use Proposition 3.2 with the finite set
E := {(1− ζkp )2p}1≤k≤ p−12
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 283
to obtain the map β1 as desired. For 1 ≤ k ≤ p−12 we have (1 − ζkp )2p = (|1 −
ζkp |ζ2k−12p )2p = |1 − ζkp |2p ∈ R so E ⊂ Q̄ ∩ R, and the set E is the Galois orbit of
(1− ζp)2p so it is fixed by Gal(Q̄/Q), hence E verifies the conditions of Proposition 3.2.
Let us denote by D(β1) the dessin corresponding to the Belyi pair (P1, β1). The Belyi
pair (P1, β1) is fixed by the action of the complex conjugation, so the embedding of D(β1)
on P1 admits a symmetry along the real line. Moreover the Belyi function β1 is a polyno-
mial, so D(β1) ∩ R is a (graph theoretic) path. Let vl < · · · < v1 be the negative vertices
on the path, and let ek denote the edge (vk−1, vk). By hypothesis β′1(0) > 0 so v1 is a
black vertex, and for k < l, the vertex vk is of even degree 2dk. We then have e
ξdk
k = ek+1
and eξ
dk
k+1 = ek if k is odd, or e
ηdk
k = ek+1 and e
ηdk
k+1 = ek if k is even. See Figure 6.
As remarked earlier, the Galois orbit of (1− ζp)2p is {(1− ζkp )2p}1≤k≤ p−12 ⊂ R−, and
(1− ζ
p−1
2
p )2p < · · · < (1− ζp)2p < 0. By construction β1((1− ζp)2p) = 0, so (1− ζp)2p
and all its Galois conjugates are black vertices of D(β1) lying on the path (v1, · · · , vl). Let
t > 0 be the index such that vt = (1− ζp)2p, and vt is a black vertex so t is odd. Then
µ0 := ξ
d1ηd2 · · · ηdt−1ξ2dtηdt−1 · · · ηd2ξd1
fixes the edge e1 (Figure 6).
vt v2 v1 ×0
e3 e2 e1
µ0
ξd1
ξd1
ηd2
ηd2
ξdt
ξdt
Figure 6: Dessin D(β1) corresponding to (P1, β1).
Let γ > 0 small enough so that β′1 > 0 on [0, γ
2pq2]. Let us next draw the dessin
D(β2) corresponding to the Belyi pair (P1, β2 = Br,s ◦Bm,n). See Figure 7.
(P1, id)0 1
(P1, Br,s)
0
r
r+s 1
r s
(P1, β2 = Br,s ◦Bm,n)0
m
m+nβ1(0) 1
s
s
mr nr
Figure 7: Dessin D(β2) corresponding to (P1, β2).
By lifting the dessin D(β2) along β1 we obtain the dessin D(β2 ◦ β1) corresponding to
the Belyi pair (P1, β2 ◦ β1). This amounts to replacing each edge of D(β1) by a copy of
D(β2). Note that the degrees of the black and white vertices are thus multiplied by mr and
284 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
nr, respectively. Analogously to µ0 we define
µ :=(ξmrd1ηξsη)(ξnrd2ηξsη) · · · (ξmrdt−2ηξsη)(ξnrdt−1ηξsη)
· (ξ2mrdtηξsη)(ξnrdt−1ηξsη)(ξmrdt−2ηξsη) · · · (ξnrd2ηξsη)ξmrd1
and we verify again that µ fixes the edge (0, v1). Note also that ξ2s fixes the edge (0, γ p
√
q).
See Figure 8.
vt v2 v1
γ2pq20
s
s
s
s
mr
mr
mr
mr
mr
mr
mr
mr
µ
nr
nr
nr
nr
η
η
η
η
η
η
ξs
ξs
ξs
ξs
ξmrd1
ξmrd1
ξnrd2
ξnrd2
ξmrdt
ξmrdt
Figure 8: Dessin D(β2 ◦ β1) corresponding to (P1, β2 ◦ β1).
Let D0 be the dessin corresponding to the Belyi pair (P1, β2 ◦ β1 ◦ β0). To simplify
the representations of the dessins we only show the vertices 0, ζk2p(1 − ζp), 1 − ζkp , and
ζk2pγ p
√
q. We decorate the vertices ζk2p(1 − ζp) (which map to (1 − ζp)2p ∈ R− by β0)
and ζk2pγ p
√
q (which map to γ2pq2 ∈ R+ by β0) respectively with the symbols ⊖ and ⊕ to
distinguish them. See Figure 9.
0
⊕
γ p
√
q
	
1− ζp
	
1− ζ̄p
⊕
⊕
	
1−
ζ 2p
	 1−
ζ
−
2
p
⊕
ζ p
γ
p√
q
⊕
ζ −
1
p
γ
p √
q
	
1
−
ζ
3 p
	
1−
ζ −
3
p
D0 = D(β2 ◦ β1 ◦ β0)
D(β2 ◦ β1) 0
γ2pq2
⊕
(1− ζp)2p
	(1− ζ2p)2p
(1− ζ3p)2p
Figure 9: Dessin D0 corresponding to (P1, β2 ◦ β1 ◦ β0).
We may now draw the Galois conjugates Dσ for σ ∈ Gal(Q̄/Q) by lifting the dessin
D0 along the projection π, by treating separately the cases σ(ζp) ∈ {ζp, ζ̄p} and σ(ζp) ∈
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 285
{ζ2p , . . . , ζp−2p }. We call the dessins respectively Dk and Djk, see Figure 10. We identify
the outermost edges on opposite sides in the representations.
?
∞
A
A
C
C
B B
D D
0 0
00
ζ•p p
√
q ζ•p p
√
q
⊕
⊕
1− ζ±1p
1− ζ±1p
	
	
⊕
	
×(2p− k)
⊕
	
×(2p− k)
⊕
	
×(k − 1)
⊕
	
×(k − 1)
µ
µ
ξ2s
ξ2s
(a) Dk := Dσ with σ(ζp) = ζ±1p .
?
∞
A
A
C
C
B B
D D
0 0
00
ζ•p p
√
q ζ•p p
√
q
⊕
⊕
1− ζjp
1− ζjp
	 	
	 	
⊕
	
×(2p− k)
⊕
	
×(2p− k)
⊕
	
×(k − 1)
⊕
	
×(k − 1)
µ
µ
ξ2s
ξ2s
(b) Djk := Dσ with σ(ζp) = ζjp where j ∈
{2, . . . , p− 2}.
Figure 10: Dessins (a) Dk and (b) Djk in the Galois orbit of D.
For all k, we have in fact D2k−1 = Dσ where σ : (ζp, p√q) 7→ (ζp, ζk−1p p
√
q), and
D2k = Dσ where σ : (ζp, p√q) 7→ (ζ̄p, ζkp p
√
q). We have similar expressions for the dessins
Djk.
Let k be fixed, and let us consider the dessin Dk. Let A denote one of the two
edges incident to 0 and on the path to the ramification point 1 − ζ±1p . We also call
B := Aη
2k−1
, C := A4p, D := Cη
2k−1
(See Figure 10a). Let E denote the set of edges inci-
dent to 0. The action of η induces the cyclic permutation of the edges of E = {Aηi}0≤i<8p.
Furthermore by construction every white vertex aside from 0 has degree 1 or 2, so η2 fixes
every edge in the complement of E. We can write E = E⊖ ⊔ E⊕ as the disjoint union
of E⊖ := {A2i}0≤i<4p and E⊕ := {B2i}0≤i<4p, such that η sends one to the other. The
action of µ on E⊖ is the transposition (A,C), and similarly the action of ξ2s on E⊕ is the
transposition (B,D).
We do the same for the dessins of the form Djk, with the only difference that this time
the action of µ on E⊖ is trivial, including on the edges A and C (see Figure 10b).
We are almost in the same configuration as in the first example. We define analogously
ω := µηµ−1ξ2sη−1µ,
and we shall prove that for some choices of γ, the actions of ω and of η2 commute only for
D1. To reproduce the proof in the first example we need only show that for some choice of
γ the actions of µηµ−1ξ2s and µ−1ξ2sη−1µ on the set E⊕ is the same as that of ξ2s.
Note that for any edge e ∈ E⊕, the edge eξi is fixed by η if i is not a multiple of s.
To that end we shall show that for some choice of γ the action of µ on E⊕ is the same as
that of ξδ , where δ is the number of occurences of ξ in the word µ, and then that δ is not a
multiple of s.
286 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
We define the words ρ1, ρ′1, ρ2, ρ
′
2, . . . , ρ2t−2, ρ
′
2t−2 ∈ F2 to be the increasing subse-
quence of the prefixes ending in η of the word µ defined above, such that ρ1 := ξmrd1η,
ρ′1 := ρ1ξ
sη, ρ2 := ρ′1ξ
nrd2η, ρ′2 := ρ2ξ
sη, etc., and µ = ρ′2t−2ξ
mrd1 . We shall show by
induction that for some choice of γ the action of ρi (resp. ρ′i) is the same as the action of
ξδi (resp. ξδ
′
i ), where δi (resp. δ′i) is the number of occurences of ξ in the word ρi (resp.
ρ′i). By induction it suffices to show that δi, δ
′
i are not multiples of s. Modulo s we have
δi ≡ δ′i equal to the non empty partial sum of
mrd1+nrd2+ · · ·+mrdt−2+nrdt−1+2mrdt+nrdt−1+mrdt−2+ · · ·+nrd2+mrd1
consisting of the first i terms.
To proceed we shall use the following result, but let us first introduce some notations.
Let P =
∑d
i=0 ciX
i ∈ Z[X] and c ∈ Z>0 such that β1 = Pc . Note that P and c do not
depend on the choice of γ, and 0 < β1(0) =
P (0)
c < 1 so 0 < c0, c− c0. We define
α := v2(γ
2pq2), ν := v2(c0) + v2(c− c0),
where v2 denotes the 2-valuation.
Lemma 3.6. If α > ν, then there exists e ∈ Z such that em ≡ c0 mod 2α and en ≡ c−c0
mod 2α, and v2(s) ≥ α− ν.
Proof. Let a, b ∈ Z coprime such that γ2pq2 = ab 2α.
Firstly,
m
m+ n
= β1(
a
b
2α) =
P (ab 2
α)
c
=
∑d
i=0 cia
i2αibd−i
bdc
,
so there exists f ∈ Z such that fm = ∑di=0 ciai2αibd−i and f(m+ n) = bdc, so
em ≡ c0 mod 2α, en ≡ c− c0 mod 2α
for e ∈ Z such that ebd ≡ f mod 2α.
Secondly,
r
r + s
= Bm,n(β1(0)) =
β1(0)
m(1− β1(0))n
β1(
a
b 2
α)m(1− β1(ab 2α))n
=
bd(m+n)cm0 (c− c0)n
(bdP (ab 2
α))m(bdc− bdP (ab 2α))n
,
so there exists g ∈ Z such that gr = bd(m+n)cm0 (c−c0)n and g(r+s) = (bdP (ab 2α))m(bdc−
bdP (ab 2
α))n. In the expansion of (bdP (ab 2
α))m, aside from the constant term bdmcm0 , ev-
ery other term is a multiple of an integer of the form ci02
αj with i ≤ m− 1 and j ≥ m− i.
By hypothesis α > ν ≥ v2(c0), so those other terms are all multiples of 2α+(m−1)v2(c0),
hence there exists A ∈ Z such that (bdP (ab 2α))m = bdmcm0 +A2α+(m−1)v2(c0). Similarly
there exists B ∈ Z such that (bdc − bdP (ab 2α))n = bdn(c − c0)n + B2α+(n−1)v2(c−c0).
Then g(r+ s) = bd(m+n)cm0 (c− c0)n +C2α+(m−1)v2(c0)+(n−1)v2(c−c0) for some C ∈ Z,
so gr = bd(m+n)cm0 (c − c0)n and gs = C2α+(m−1)v2(c0)+(n−1)v2(c−c0). The integers r
and s are coprime, so after dividing gr and gs by their greatest common dividor we obtain
that
v2(s) ≥ α− ν > 0.
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 287
Using this lemma, we know that if α > ν, then there exists e ∈ Z such that em ≡ c0
mod 2α and en ≡ c − c0 mod 2α, v2(s) ≥ α − ν where ν does not depend on γ, and r
is coprime to s so is not a multiple of 2. Therefore there exists e′ ∈ Z such that e′mr ≡ c0
mod 2α and e′nr ≡ c − c0 mod 2α. Moreover 2α−ν is a common divisor of 2α and s,
so by the above modulo 2α−ν we have e′δi ≡ e′δ′i equal to the non empty partial sum δ̃i
consisting of the first i terms of the sum
c0d1 + (c− c0)d2 + · · ·+ c0dt−2 + (c− c0)dt−1 + 2c0dt
+(c− c0)dt−1 + c0dt−2 + · · ·+ (c− c0)d2 + c0d1.
Similarly e′δ is equal modulo 2α−ν to the whole sum
δ̃ := 2(c0d1 + (c− c0)d2 + · · ·+ c0dt−2 + (c− c0)dt−1 + c0dt).
By construction c0, c − c0, di are positive and do not depend on the choice of γ, so
0 < c0d1 ≤ δ̃i ≤ δ̃, thus for any choice of γ such that α > ν and δ̃ < 2α−ν (for instance
γ = 2
u
2v+1 with 1 ≪ u ≪ v), we obtain δ̃i, δ̃ ̸≡ 0 mod 2α−ν , and in consequence δi, δ′i
and δ are not multiples of s. Therefore we can now conclude by induction that the actions
of ρi and ρ′i are the same as that of ξ
δi and ξδ
′
i , respectively. Indeed, δ1 is not a multiple
of s so ρ1 = ξδ1η and ξδ1 have the same action on E⊕. If ρi has the same action as ξδi
on E⊕, then ρ′i = ρiξ
sη has the same action as ξδiξsη = ξδ
′
iη on E⊕, and also the same
action as ξδ
′
i because δ′i is not a multiple of s. Similarly, if ρ
′
i has the same action as ξ
δ′i
on E⊕, then ρi+1 has the same action as ξδi+1η on E⊕, and also the same action as ξδi+1
because δi+1 is not a multiple of s.
We have thus proved that µ has the same action as ξδ on E⊕, and by symmetry µ−1
has the same action as ξ−δ on E⊕. And δ and 2s− δ are not multiples of s, so µηµ−1ξ2s
and µ−1ξ2sη−1µ have the same action as ξ2s on E⊕, as announced. We shall now observe
the action of ω = µηµ−1ξ2sη−1µ on E. Let Mk and M
j
k denote the monodromy maps of
the dessins Dk and Djk.
For the dessins Dk for 1 ≤ k ≤ 2p, the action of µ on E⊖ is the transposition (A,C),
and the action of ξ2s on E⊕ is the transposition (B,D), therefore the action of ω fixes the
set E on which it induces the permutation
Mk(ω)|E = (A,C)(B,D) ·
4p−1∏
i=0
(Aη
2i
, Aη
2i+1
) · (A,C)(B,D).
Hence for k = 1,
M1(ω)|E = (A,Aη
4p
)(Aη, Aη
4p+1
) ·
4p−1∏
i=0
(Aη
2i
, Aη
2i+1
) · (A,Aη4p)(Aη, Aη4p+1)
=
4p−1∏
i=0
(Aη
2i
, Aη
2i+1
)
so ω and η2 commute on E. Moreover η2 acts trivially on the complement of E, so finally
M1(ω) and M1(η2) commute.
288 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
For k = 2, we observe that Bωη
2
= Dη
−1η2 = Dη but Bη
2ω = Bη
2η−1 = Bη .
Similarly, for 3 ≤ k ≤ 2p, we observe that Aωη2 = Cηη2 = Cη3 but Aη2ω = Aη2η = Aη3 .
Therefore Mk(ω) and Mk(η2) do not commute for 2 ≤ k ≤ 2p.
For the dessins Djk for 1 ≤ k ≤ 2p and 2 ≤ j ≤ p−12 , ξ2s on E⊕ is the transposition
(B,D), and µ acts trivially on E⊖, therefore the action of ω fixes the set E on which it
induces the permutation
M jk(ω)|E = (B,D) ·
4p−1∏
i=0
(Aη
2i
, Aη
2i+1
) · (B,D).
Hence we observe that Bωη
2
= Dη
−1η2 = Dη but Bη
2ω = Bη
2η−1 = Bη , so M jk(ω) and
M jk(η
2) do not commute.
We have thus shown that the actions of ω and η2 commute only for D1, this concludes
the proof that D̃ is a regular dessin with moduli field Q(ζp, p√q).
3.3 Regular dessin with moduli field Q(ζ3, 3
√
3)
Finally, let us exhibit a regular dessin with moduli field Q(ζ3, 3
√
3) of smaller degree by
choosing a Belyi map that is a rational function instead of a polynomial as was done in the
previous subsections. Let
C : y2 = x(x− (1− ζ3))(x− 3
√
3),
β : C → P1, (x, y) 7→ (x+ 3
3)3
35(x− 32)2 .
The function β is given by the composition of the following maps β = β1 ◦ β0 ◦ π.
1. π : C → P1 is the projection on the coordinate x, which ramifies over {0, 1 −
ζ3,
3
√
3,∞}.
2. β0 := X6 ∈ Q[X], Crit(β0) = {0} so β0 ◦ π ramifies over {0, (1 − ζ3)6 =
−33, 32,∞}.
3. β1 :=
(X+33)3
35(X−32)2 , Crit(β1) = {0, 1} so β = β1 ◦ β0 ◦ π ramifies over {0, 1,∞}.
The pair (C, β) is thus a Belyi pair, and we call D the dessin corresponding to (C, β).
Similarly as in 3.1, D has moduli field Q(ζ3, 3
√
3). We will proceed analogously to show
that the regular closure D̃ has the same field of moduli. Let us first draw the dessin D0
corresponding to the Belyi pair (P1, β1 ◦ β0) (see Figure 11), and lift it to the conjugate
curves Cσ to obtain the conjugate dessins Dσ for σ ∈ Gal(Q(ζ3, 3
√
3)/Q) (see Figure 12).
As usual we identify the outermost edges on opposite sides.
We can now compute the cartographic groups of the dessins. Let Mk denote the mon-
odromy map of Dk. Then
Mk(ξ) =
(1,13,14,7,25,26)(2,15,16)(3,17,18)(4,19,20)(5,21,22)
(6,23,24)(8,27,28)(9,29,30)(10,31,32)(11,33,34)(12,35,36)
for all 1 ≤ k ≤ 6, and
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 289
(P1, id)0 1
(P1, β1)?
−33 0 1 32
(P1, β1 ◦ β0)
0
?
??
?
? ?
1− ζ23
1− ζ3
3
√
3
ζ3
3
√
3
ζ23
3
√
3
Figure 11: Construction of D0.
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13
14
25
26
0 0
00
∞3√3 3
√
3
1− ζ3
1− ζ3
1 7
1 7
(a) D1 := D
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13
14
25
26
0 0
00
∞
ζ3
3
√
3 ζ3
3
√
3
1− ζ23
1− ζ23
1 7
1 7
(b) D2 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , ζ3 3
√
q)
290 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13 14
25
26
0 0
00
∞
ζ3
3
√
3 ζ3
3
√
3
1− ζ3
1− ζ3
1 7
1 7
(c) D3 := Dσ, σ : (ζ3, 3√q) 7→ (ζ3, ζ3 3√q)
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13
14
2526
0 0
00
∞
ζ23
3
√
3 ζ23
3
√
3
1− ζ23
1− ζ23
1 7
1 7
(d) D4 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , ζ23 3
√
q)
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13
14
25
26
0 0
00
∞
ζ23
3
√
3 ζ23
3
√
3
1− ζ3
1− ζ3
1 7
1 7
(e) D5 := Dσ, σ : (ζ3, 3√q) 7→ (ζ3, ζ23 3
√
q)
? ??
2
8
3
9
4
10
5
11
6
12
15
27
16
28
17
29
18
30
19
31
20
32
21
33
22
34
23
35
24
36
13
14
25
26
0 0
00
∞3√3 3
√
3
1− ζ23
1− ζ23
1 7
1 7
(f) D6 := Dσ, σ : (ζ3, 3√q) 7→ (ζ23 , 3
√
q)
Figure 12: Dessins D1, . . . ,D6 in the Galois orbit of D.
N. Daire et al.: Regular dessins with moduli fields of the form Q(ζp, p
√
q) 291
• M1(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,36)(14,15)(16,17)(18,19)
(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35) ,
• M2(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,36)(14,27)(15,26)(16,29)
(17,28)(18,19)(20,21)(22,23)(24,25)(30,31)(32,33)(34,35) ,
• M3(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,36)(14,27)(15,26)(16,17)
(18,31)(19,30)(20,21)(22,23)(24,25)(28,29)(32,33)(34,35) ,
• M4(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,36)(14,27)(15,26)(16,17)
(18,19)(20,33)(21,32)(22,23)(24,25)(28,29)(30,31)(34,35) ,
• M5(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,36)(14,27)(15,26)(16,17)
(18,19)(20,21)(22,35)(23,34)(24,25)(28,29)(30,31)(32,33) ,
• M6(η) =
(1,2,3,4,5,6,7,8,9,10,11,12)(13,24)(14,27)(15,26)(16,17)
(18,19)(20,21)(22,23)(25,36)(28,29)(30,31)(32,33)(34,35) .
Using the computer algebra system SageMath [10], we determined that
|⟨M1(ξ),M1(η)⟩| = 42467328 = 219 · 34.
Moreover, M1(ξ), M1(η) and M1(ξη) respectively have orders 6, 12 and 12, so the Euler
characteristic of the underlying surface of D̃1 is
χ = |⟨M1(ξ),M1(η)⟩| · (
1
ordM1(ξ)
+
1
ordM1(η)
+
1
ordM1(ξη)
− 1)
= −28311552 = −220 · 33,
and its genus is g = 1− χ2 = 14155777.
We will now show that D̃1 is not isomorphic to D̃2, . . . , D̃6. We claim that ω :=
[ξ−1η2ξ, ξη] ∈ kerM1 \
⋃
2≤k≤6 kerMk, thus concluding the proof. Indeed, we obtain:
• M1(ω) = id;
• M2(ω) = (13, 25)(15, 27)(21, 33)(23, 35);
• M3(ω) = (17, 29)(21, 33);
• M4(ω) = (13, 25)(15, 27)(19, 31)(21, 33);
• M5(ω) = (13, 25)(17, 29);
• M6(ω) = (13, 25)(19, 31)(21, 33)(23, 35).
We have thus constructed a regular dessin D̃ of degree 219 · 34 and genus 14155777
with moduli field Q(ζ3, 3
√
3).
ORCID iDs
Fumiharu Kato https://orcid.org/0009-0002-4800-0029
292 Ars Math. Contemp. 24 (2024) #P2.05 / 273–292
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.06 / 293–315
https://doi.org/10.26493/1855-3974.2619.06c
(Also available at http://amc-journal.eu)
Generalized X-join of graphs and their
automorphisms*
Javad Bagherian †, Hanieh Memarzadeh
Department of Pure Mathematics, Faculty of Mathematics and Statistics,
University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran
Received 4 May 2021, accepted 6 April 2023, published online 27 September 2023
Abstract
In this paper, we first introduce a new product of finite graphs as a generalization of
the X-join of graphs. We then give necessary and sufficient conditions for a graph to be
isomorphic to a generalized X-join. As a main result, we give necessary and sufficient
conditions under which the full automorphism group of a generalized X-join is equal to the
generalized wreath product of the automorphism groups of its factors.
Keywords: Automorphism, generalized wreath product, graph, lexicographic product, permutation
group, X-join.
Math. Subj. Class. (2020): 05C25, 20B25, 20E22
1 Introduction
One of the main problems in the theory of graphs, known as the König problem, asks for a
concrete characterization of all automorphism groups of graphs. In particular, the problem
of computing a generating set of the automorphism group is equivalent to the graph isomor-
phism problem [9]. The automorphism groups of many graphs can be expressed in terms of
the automorphism groups of their subgraphs. For instance, in most cases the automorphism
groups of the graphs which are the lexicographic product of graphs are expressed in terms
of the automorphism groups of their factors. The lexicographic product of graphs is one of
the important products of graphs, defined by Harary in [7]. Sabidussi in [11] showed that
under some conditions the automorphism group of the lexicographic product of two graphs
*The authors are grateful to the anonymous referees, whose comprehensive reports helped to improve the
quality of this paper.
†Corresponding author.
E-mail addresses: bagherian@sci.ui.ac.ir (Javad Bagherian), h.memarzadeh.762@sci.ui.ac.ir (Hanieh
Memarzadeh)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
294 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
Γ and Γ′ can be expressed as the wreath product of the automorphism groups of Γ and Γ′.
An important generalization of the lexicographic product is the X-join. It was introduced
by Sabidussi as the graph formed from a given graph Γ = (V,R) by replacing every vertex
v of Γ by a graph Bv and joining the vertices of Bv with those of Bu whenever uv ∈ R
[11]. Note that the graphs Bv , v ∈ V , need not be mutually isomorphic. Hemminger in
[8] gave necessary and sufficient conditions for the automorphism group of the X-join of
graphs {Bv}v∈V to be the natural ones, i.e., those that are obtained by first permuting the
graphs Bv , v ∈ V , according to a permutation of subscripts by an automorphism of Γ
and then performing an arbitrary automorphism of each Bv . Note that Hemminger did not
determine the structure of the automorphism group of the X-join of {Bv}v∈V in terms of
automorphism groups of Bv , v ∈ V . It should be mentioned that the above results have
been generalized to directed color graphs in [3]. If for a color digraph C = (V,R) and a
collection of color digraphs {Dc | c ∈ V }, each vertex c of C is replaced by a copy of Dc
and all possible arcs of color k from Dc to Dc′ are included, if and only if there is an arc
of color k from c to c′ in C, we get the C-join of these color digraphs. The wreath product
of two color digraphs C and D is the C-join of {Dc | c ∈ V } where Dc ∼= D for every
c ∈ V . In [3], all automorphism groups of digraphs that can be written as a wreath product
have been determined.
In this paper we first give a generalization of the X-join of graphs (see Definition 2.1).
This generalization, as a new operation on finite graphs, is a natural generalization of the
X-join of graphs (a more algebraic way was considered by Weisfeiler [12, page 45] as
the wreath product of a family of stable graphs with another stable graph). Also this new
graph product generalizes the generalized wreath product of circulant digraph which de-
fined in [2] (see Remark 2.9). It is also closely related with the wedge product of association
schemes introduced and studied in [10] (see Remark 2.8). In Section 2 we give necessary
and sufficient conditions under which a graph is isomorphic to a generalized X-join (see
Theorem 2.4). But the main result of this paper deals with the connections between the au-
tomorphism group of a generalized X-join and the automorphism groups of its factors. For
computing the automorphism group of the generalized X-join of graphs, we need a general-
ization of the wreath product of permutation groups. Recently, such a generalization, called
the generalized wreath product, has been given in [1, 5]. We first show that under some
conditions the automorphism group of the generalized X-join of graphs contains the gen-
eralized wreath product of the automorphism groups of their factors (Theorem 4.1). As a
main result, we then give necessary and sufficient conditions under which the full automor-
phism group of the generalized X-join of graphs is equal to the generalized wreath product
of the automorphism groups of their factors (Theorem 4.2). In particular, we determine the
structure of the natural automorphism group of the X-join of graphs (Corollary 4.7).
Terminology and notation: Throughout this paper, by a graph Γ = (V,R) we mean
a finite undirected graph without multiple edges with the vertex set V = V (Γ) and the
edge set R = E(Γ). We denote the complement of Γ by Γ. If all pairs of vertices of
a subgraph Γ′ of Γ that are adjacent in Γ are also adjacent in Γ′, then Γ′ is an induced
subgraph. For X ⊆ V we write Γ[X] for the subgraph of Γ induced by X and we also
denote by Γ(X) the graph with vertices X and edge set E(Γ[X]) ∪ {(x, x) | x ∈ X}.
For two graphs Γ = (V,R) and Γ′ = (V ′, R′), by a graph homomorphism f : Γ → Γ′ we
mean a mapping f : V → V ′ such that (f(u), f(v)) ∈ R′ whenever (u, v) ∈ R. In the
case when f : V → V ′ is surjective, f : Γ → Γ′ is called a graph epimorphism. More-
J. Bagherian: Generalized X-join of graphs and their automorphisms 295
over, if f : V → V ′ is a bijection and f−1 : Γ′ → Γ is also a graph homomorphism, then
f : Γ → Γ′ is called a graph isomorphism. Two graphs Γ and Γ′ are called isomorphic if
there exists a graph isomorphism between Γ and Γ′. In this case we write Γ ≃ Γ′. When
Γ = Γ′ every graph isomorphism f : Γ → Γ is called a graph automorphism of Γ. The
set of all graph automorphisms of Γ is denoted by Aut(Γ) and is called the automorphism
group of Γ.
If Π is a partition of the vertices of a graph Γ, then the quotient graph Γ/Π is a graph
with vertex set Π, for which distinct classes X,X ′ ∈ Π are adjacent if some vertex in X is
adjacent to a vertex of X ′.
Let Γ = (V,R) be a graph. The X-join of a set of graphs {Bx = (Yx, Ex) | x ∈ V }
with Γ, denoted by Γ[Bx]x∈V , is a graph W = (Y,E) where Y =
⋃̇
x∈V Yx and
E = {(yx, y′x′) ∈ Yx × Yx′ | (x, x′) ∈ R, or else x = x′ and (yx, y′x) ∈ Ex}.
If B = (Y ′, E′) and Bx = B for every x ∈ V , we can identify
⋃̇
x∈V Yx with Y
′ × V and
then the X-join of {Bx = (Yx, Ex) | x ∈ V } is the lexicographic product of Γ and B and
is denoted by Γ ◦B.
We denote by Kn a complete graph with n vertices. For the graph theoretical terminol-
ogy and notation that are not defined here, we refer the reader to [6].
For a finite set V , we denote by Sym(V ) the group of all permutations of V . Every
subgroup of Sym(V ) is called a permutation group on V . For F ≤ Sym(V ) and ∆ ⊆ V ,
the setwise stabilizer of ∆ in F is F{∆} = {f ∈ F | ∆f = ∆} and the pointwise stabilizer
of ∆ in F is F(∆) = {f ∈ F | xf = x, ∀x ∈ ∆}. We say that two permutation groups
F ≤ Sym(V ) and F ′ ≤ Sym(V ′) are permutation isomorphic if there exist a bijection
λ : V → V ′ and a group isomorphism η : F → F ′ such that for every f ∈ F and v ∈ V
we have λ(vf ) = λ(v)η(f).
By a system of blocks Π for a permutation group F ≤ Sym(Ω) we mean
(1) Π is a partition of Ω;
(2) for every ∆ ∈ Π and every f ∈ F , ∆f ∩∆ = ∅ or ∆f = ∆.
If Π is a system of blocks for F and ∆ ∈ Π, by F∆ we mean the group induced by the
action of F{∆} on ∆. Then F∆/F(∆) ≤ Sym(∆) is a permutation group.
2 A generalization of the X-join of graphs
In this section we first introduce a new product of graphs, called the generalized X-join of
graphs. Then we give necessary and sufficient conditions under which a graph is isomor-
phic to a generalized X-join.
Definition 2.1. Let Γ = (V,R) be a graph and Π be a partition of V . Suppose that for every
X ∈ Π we are given a graph BX = (YX , EX) and a graph epimorphism πX : YX → X
from BX onto Γ(X). Put Y =
⋃̇
X∈ΠYX and π =
⋃̇
X∈ΠπX where for every y ∈ YX ,
π(y) := πX(y). We define a graph W with vertex set Y and edge set E such that (y, y′) ∈
E if and only if
(1) either (y, y′) ∈ EX , for some X ∈ Π;
(2) or (y, y′) ∈ π−1X (x)× π
−1
X′ (x
′) where X ̸= X ′ and (x, x′) ∈ R.
296 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
We call the graph W = (Y,E) the generalized X-join of Γ and {BX}X∈Π with respect to
π, and we denote it by Γ ◦π {BX}X∈Π. (See Figure 1.)
Figure 1: The generalized X-join of Γ and {BX}X∈Π.
In the following we show that the X-join of graphs is a special case of the generalized
X-join of graphs.
Example 2.2. Let Γ = (V,R) be a graph and Π be a partition of V such that for every
X ∈ Π, X = {x} for some x ∈ V . Suppose that {Bx = (Yx, Ex) | x ∈ V } is a
set of graphs. Define a graph epimorphism πx : Yx → X from Bx onto Γ(X) such that
πx(yx) = x for every yx ∈ Yx. Then the generalized X-join of Γ and {Bx}x∈Π with
respect to π =
⋃̇
x∈V πx is a graph with vertices Y =
⋃̇
x∈V Yx and the edge set E such
that (yx, y′x′) ∈ E if and only if
(1) either x = x′ and (yx, y′x) ∈ Ex;
(2) or x ̸= x′ and (x, x′) ∈ R.
One can see that in this case Γ ◦π {Bx}x∈V = Γ[Bx]x∈V , the X-join of graphs {Bx}x∈V .
Example 2.3. Let Γ = (V,R) be the graph in Figure 2. Consider the partition Π =
{X,X ′, X ′′} of V where X = {1, 2}, X ′ = {3, 4}, and X ′′ = {5, 6}. Suppose that
BX = (YX , EX), BX′ = (YX′ , EX′), and BX′′ = (YX′′ , EX′′) are the graphs in Figure 2
with vertices YX = {a, b, c}, YX′ = {d, e, f}, and YX′′ = {g, h, i, k}, respectively.
Now define the graph epimorphisms πX : BX → Γ(X), πX′ : BX′ → Γ(X ′), and
πX′′ : BX′′ → Γ(X ′′) as follows:{
πX(a) = πX(b) = 1
πX(c) = 2{
πX′(e) = πX′(d) = 3
πX′(f) = 4
J. Bagherian: Generalized X-join of graphs and their automorphisms 297
2
1 3 4 6
5 Γ c
b
a
BX
e
d
f
BX′
hg
i k BX′′
Figure 2: Graph Γ and set of graphs {BX}X∈Π.
{
πX′′(g) = πX′′(h) = 5
πX′′(i) = πX′′(k) = 6.
Then the generalized X-join of Γ and {BX , BX′ , BX′′} with respect to π is the graph
in Figure 3.
c
b
a
e
d
f
h
g
i
k
Figure 3: Graph W = Γ ◦π {BX , BX′ , BX′′}.
Let Γ = (V,R) be a graph and let A,B ⊆ V . We say that A is externally related with
respect to B, if every vertex v ∈ B that is adjacent to at least one element in A is adjacent
to all vertices of A. Moreover, if B is also externally related with respect to A, we say that
A and B are externally related to each other.
Suppose that W = (Y,E) is the generalized X-join of Γ = (V,R) and {BX =
(YX , EX) | X ∈ Π} with respect to π. Then we can define two equivalence relations
E0 and E1 on Y as follows:
(u, v) ∈ E0 ⇔ u, v ∈ π−1X (x), for some X ∈ Π and x ∈ X; (2.1)
(u, v) ∈ E1 ⇔ u, v ∈ YX , for some X ∈ Π. (2.2)
Clearly, E0 ⊆ E1. In the following we give a characterization of the generalized X-join of
graphs in terms of the equivalence relations E0 and E1.
Theorem 2.4. A graph W = (Y,E) is a generalized X-join of graphs if and only if there
exist two equivalence relations E0 and E1 on Y such that
298 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
(i) E0 ⊆ E1;
(ii) for every equivalence class P of E0 which is contained in a equivalence class Q of
E1, P is externally related with respect to every equivalence class of E0 which is not
in Q.
Proof. Suppose that W = (Y,E) is the generalized X-join of Γ and {BX}X∈Π with re-
spect to π. Then as we saw above, there are two equivalence relations E0 and E1 on Y
such that E0 ⊆ E1. Since for every x ∈ X and x′ ∈ X ′ where X ̸= X ′, π−1X (x) and
π−1X′ (x
′) are externally related to each other, it follows that condition (ii) holds.
Now suppose that there exist two equivalence relations E0 and E1 on Y such that
conditions (i) and (ii) hold. Let Y/E0 and Y/E1 be the sets of the equivalence classes of
E0 and E1 on Y , respectively. Let Γ be the quotient graph W/E0. Moreover, for every
U ∈ Y/E1, let U0 be the equivalence classes of E0 which are contained in U and BU0 be
the subgraph of W induced by U . Since E0 ⊆ E1, {U0 | U ∈ Y/E1} gives a partition
Π on Y/E0. Then for every U ∈ Y/E1 we can define a graph epimorphism πU0 from the
graph BU0 onto Γ(U0). Suppose that W
′ is the generalized X-join of Γ and {BU0}U0∈Π
with respect to π. Then V (W ′) = Y and it follows from condition (ii) that the set of edges
ofW andW ′ are the same. ThusW =W ′ and soW is a generalized X-join of graphs.
Remark 2.5. The following example shows that unlike the X-join of graphs, a graph can
be represented as a generalized X-join of graphs, but not a unique way. This means that if
W and W ′ are two isomorphic generalized X-join of graphs then it is not necessarily true
that the factors of W and W ′ are isomorphic.
Example 2.6. Consider the graph W = (Y,E) in Figure 4. If we consider two equiva-
lence relations E0 ⊆ E1 such that Y/E0 = {{a}, {b, c}, {d, e, f, g}, {h}} and Y/E1 =
{{a, b, c}, {d, e, f, g, h}}, then one can see that conditions (i) and (ii) of Theorem 2.4
hold. So it follows from Theorem 2.4 that W = Γ ◦π {BX , BX′} where the graphs Γ, BX
and BX′ are shown in Figure 5. On the other hand, consider the graphs Γ′, B′X and B
′
X′
that are shown in Figure 6. Set Π′ = {X = {1, 2, 3}, X ′ = {4, 5}} and define the graph
epimorphisms π′X : B
′
X → Γ′(X) by 
a −→ 1
b, c −→ 2
h −→ 3
and π′X′ : B
′
X′ → Γ′(X ′) by {
d, e −→ 4
f, g −→ 5
.
Then one can see that W is the generalized X-join Γ′ ◦π′ {B′X , B′X′} with respect to π′.
The following lemma that gives a sufficient condition under which two generalized
X-join are isomorphic, is straightforward and therefore left to the reader.
Lemma 2.7. Let W = Γ ◦π {BX}X∈Π and W ′ = Γ′ ◦π′ {BX′}X′∈Π′ . Suppose that the
following conditions hold.
J. Bagherian: Generalized X-join of graphs and their automorphisms 299
a
b
h
c
d
f g
e
Figure 4: Graph W .
2
1
3
4 Γ
a
b c BX
d
f g
e
h BX′
Figure 5: Graph Γ and set of graphs {BX , BX′}.
(1) There exists a graph isomorphism α : Γ → Γ′ which maps every partition class
X ∈ Π onto a partition class X ′ ∈ Π′;
(2) For every X ∈ Π, there exist graph isomorphisms βXX′ : BX → BX′ with X ′ =
Xα such that the following diagram is commutative.
YX
βXX′−−−−→ YX′
πX
y yπX′
X
α−−−−→ X ′
Then ψ :
⋃̇
X∈ΠYX →
⋃̇
X′∈Π′YX′ defined by yX → βXX′(yX) is a graph isomorphism
between W and W ′.
Remark 2.8. The generalized X-join is closely related with the wedge product of asso-
ciation schemes. The wedge product of association schemes which provides a way to
construct new association schemes from old ones has been given in [10]. In the following
we give the relationship between the relations of a wedge product of symmetric association
schemes and the generalized X-join.
300 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
1
2
3
5
4
Γ′
a
b
h
c
B′X
d
f g
e
B′X′
Figure 6: Graph Γ′ and set of graphs {B′X , B′X′}.
Suppose (V,G) is an association scheme and E is an equivalence relation on V such
that it is a union of some relations R0, R1, . . . , Rt of G. Put D = {R0, R1, . . . , Rt} and
suppose Σ is the set of equivalence classes of E. For every X ∈ Σ, let DX = {gX | g ∈
D} where gX = g ∩X ×X . Moreover, assume that
(1) there is a set of association schemes {(YX , BX) | X ∈ Σ} such that all YX are
pairwise disjoint and for every X ∈ Σ there exists a scheme normal epimorphism
πX : YX ∪BX → X ∪DX .
(2) for every X,X ′ ∈ Σ, there exists an algebraic isomorphism φXX′ : BX → BX′
such that the diagram
BX
φXX′−−−−→ BX′
πX
y yπX′
DX
εXX′−−−−→ DX′
is commutative, where εXX′(gX) = gX′ .
Put Y :=
⋃̇
X∈ΣYX , π :=
⋃̇
X∈ΣπX and for every b ∈ BX , b̃ =
⋃
X′∈Σ φXX′(bX).
Moreover, for every g ∈ G put
g =
⋃
(x,x′)∈g∩X×X′,
X,X′∈Σ,X ̸=X′
ψ−1X (x)× ψ
−1
X′ (x
′).
Fix Z ∈ Σ. Put B̃Z = {b̃ | b ∈ BZ}. Then it follows from [10, Theorem 2.2] that
the pair (Y, B̃Z ∪ (G \ D)) is an association scheme, is called the wedge product of
(YX , BX), X ∈ Σ, and (V,G). Now let g ∈ G\D and bX ∈ BX such that πX(bX) = gX .
Then one can see that the graph with vertices Y and the edge set g ∪ b̃ is the generalized
X-join of g and {φXX′(bX)}X′∈Σ with respect to π.
Remark 2.9. The generalized wreath product of Cayley digraphs on abelain groups was
first introduced in [2] and an entire section of the recent book [4, Section 5] is devoted
to their study. A Cayley digraph Cay(G,S) of G with connection set S is a generalized
wreath product if there are subgroups 1 < K ≤ L < G such that S \L is a union of cosets
of K. In the following we show that the generalized X-join generalizes the generalized
wreath product. To see this, let Cay(G,S) be a generalized wreath product on abelian
J. Bagherian: Generalized X-join of graphs and their automorphisms 301
group G such that 1 ̸∈ S and S = S−1. Let V = {g1, . . . , gt} be a set of left coset
representatives of K in G and Γ be the subgraph of G induced on V . Suppose that {a0 =
1, a1, . . . , am} is a set of left coset representatives of L in G. For every 0 ≤ i ≤ m, let
Xi = {gj ∈ V | gj ∈ aiL}. Then Π = {X0, X1, . . . , Xm} is a partition of V . Put
B0 = Cay(L,L ∩ S) and for every 1 ≤ i ≤ m, let Bi = ϕi(B0) where ϕi : B0 → Bi
is a graph isomorphism defined by ϕi(l) = ail for every l ∈ L. Then there is the graph
epimorphism πi : Bi → Γ(Xi) such that πi(gjK) = gj . Now let W = (G,E) be the
generalized X-join of Γ and {B0, B1, . . . , Bm} with respect to π where π =
⋃̇m
i=0πi. We
show that for every x, y ∈ G, xy ∈ E if and only if xy−1 ∈ S. Clearly, if x, y ∈ aiL,
then xy ∈ E if and only if xy−1 ∈ S ∩ L. If x ∈ grK ⊆ aiL and y ∈ gsK ⊆ ajL where
i ̸= j, then xy−1 ∈ grg−1s K. Then xy ∈ E if and only if grg−1s ∈ S \ L if and only if
grg
−1
s K ⊆ S \ L, since S \ L is a union of cosets of K. So in this case xy ∈ E if and
only if xy−1 ∈ S \L. Thus we conclude that W = (G,E) = Cay(G,S). This means that
Cay(G,S) is a generalized X-join.
3 Generalized wreath product, definition and construction
The generalized wreath product of permutation groups has been defined in [1, 5]. Since in
the next section we need to construct the generalized wreath product of the automorphism
group of graphs, here we have a look at the definition of this product which has been given
in [1].
Let Γ = (V,R) be a graph and F = Aut(Γ). Suppose that Π is a system of blocks for
F . Moreover, suppose that we are given a set of graphs {BX = (YX , EX) | X ∈ Π} such
that the following conditions hold.
(G1) If for some f ∈ F , Xf = X ′, then BX ≃ BX′ ,
(G2) If ∆ is an orbit of F on Π, then for some X ∈ ∆, there exists a graph epimorphism
πX : YX → X fromBX onto Γ(X) and there exists an epimorphism ηX : Aut(BX)
→ FX/F(X) such that
πX(y
l) = (πX(y))
ηX(l), ∀y ∈ YX , l ∈ Aut(BX).
By condition (G1), if there exists fXX′ ∈ F such that XfXX′ = X ′, we have a graph
isomorphism ϕXX′ : YX → YX′ from graph BX onto BX′ . Then ψXX′ : Aut(BX) →
Aut(BX′) defined by
ψXX′(α) = ϕXX′αϕ
−1
XX′ , ∀ α ∈ Aut(BX),
is an isomorphism from Aut(BX) onto Aut(BX′).
Moreover, by condition (G2), ΛX = {π−1X (x) | x ∈ X} is a system of blocks for
Aut(BX),
ηX : Aut(BX)/KX → FX/F(X)
is an isomorphism, and Aut(BX)/KX ≤ Sym(ΛX) and FX/F(X) ≤ Sym(X) are per-
mutation isomorphic where KX = ker(ηX).
Lemma 3.1. Let X ′ ∈ ∆ with X ′ ̸= X . Then
302 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
(1) there exists a graph epimorphism πX′ from BX′ onto Γ(X ′) such that the following
diagram is commutative, where ξXX′ : X → X ′, given by ξXX′(x) = xfXX′ for
every x ∈ X.
YX
ϕXX′−−−−→ YX′
πX
y yπX′
X
ξXX′−−−−→ X ′
(2) there exists an epimorphism ηX′ : Aut(BX′) → FX
′
/F(X′) such that the follow-
ing diagram is commutative, where ρXX′ : FX/F(X) → FX
′
/F(X′), defined by
ρXX′(hF(X)) = fXX′hf
−1
XX′F(X′) for every hF(X) ∈ FX/F(X).
Aut(BX)
ψXX′−−−−→ Aut(BX′)
ηX
y yηX′
FX/F(X)
ρXX′−−−−→ FX′/F(X′)
Proof. (1) If we define πX′ = ξXX′πXϕ−1XX′ , then πX′ : YX′ → X ′ is a graph epimor-
phism fromBX′ onto Γ(X ′) such that the diagram mentioned above is commutative.
(2) Define ηX′ = ρXX′ηXψ−1XX′ . Then ηX′ : Aut(BX′) → FX
′
/F(X′) is an epimor-
phism such that the above diagram is commutative.
Now suppose that a graph Γ = (V,R) and a set of graphs {BX = (YX , EX) | X ∈ Π}
satisfy conditions (G1) and (G2). Set Y =
⋃̇
X∈ΠYX . Since for every X ∈ Π, KX ≤
Aut(BX) it follows that the action of
∏
X∈ΠKX on Y defined by
yk := ykX , y ∈ YX , k =
∏
X∈Π
kX ∈ K
is faithful. Set K =
∏
X∈ΠKX . Then K ≤ Sym(Y ).
Moreover, every element of F can be also considered as an element of Sym(Y ). In
fact, for every g ∈ F we can associate g ∈ Sym(Y ). To do this, let yX ∈ YX and TX be a
set of left coset representatives for KX in Aut(BX) such that idYX ∈ TX . Let g ∈ F . We
associate to g an element g ∈ Sym(Y ) as follows:
(i) if Xg = X , then (yX)g = (yX)t where ηX(tKX) = gF(X) for some t ∈ TX ;
(ii) if Xg = X ′, then (yX)g = ϕXX′((yX)t) where ηX(tKX) = f−1XX′gF(X) for some
t ∈ TX .
Set F = {g | g ∈ F}. Clearly, F ⊆ Sym(Y ) and ⟨K,F ⟩ ≤ Sym(Y ). According to
[1, Definition 2.1], the permutation group ⟨K,F ⟩, is the generalized wreath product of
{Aut(BX)}X∈Π and F . We denote it by F ◦ {Aut(BX)}X∈Π.
Remark 3.2. It should be mentioned that the generalized wreath product of {Aut(BX)}X∈Π
and F is independent of the choice of representatives TX for every X ∈ Π. To see this,
let yX ∈ YX and T ′X be a set of left coset representatives for KX in Aut(BX) such that
J. Bagherian: Generalized X-join of graphs and their automorphisms 303
idYX ∈ T ′X and T ′X ̸= TX . Let g ∈ F and ĝ be an element of Sym(Y ) associated with g
by the above argument. If Xg = X , then (yX)ĝ = (yX)t
′
where ηX(t′KX) = gF(X) for
some t′ ∈ T ′X . Since t′ = tkX , for some t ∈ TX and kX ∈ KX , we have
(yX)
ĝ = (yX)
t′ = (yX)
tkX = (yX)
gkX .
Similarly, if Xg = X ′, then (yX)ĝ = ϕXX′((yX)t
′
) where ηX(t′KX) = f−1XX′gF(X) for
some t′ ∈ T ′X . If t′ = tkX for some t ∈ TX and kX ∈ KX , then
(yX)
ĝ = ϕXX′((yX)
t′) = ϕXX′((yX)
tkX ) = ϕXX′((yX)
gkX ).
Then we conclude that
< F̂ ,K >=< F,K >,
where F̂ = {f̂ | f ∈ F}. This shows that F ◦ {Aut(BX)}X∈Π =< F̂ ,K >.
Example 3.3. Let Γ and {BX , BX′ , BX′′} be the graphs in Figure 7. Then
Aut(Γ) = {idV , (14)(25)(36), (23), (56), (23)(56), (2635)(14), (2536)(14),
(26)(35)(14)}
and
Π = {X = {2, 3}, X ′ = {5, 6}, X ′′ = {1, 4}}
is a system of blocks for F = Aut(Γ). Put fXX′ = (14)(25)(36). Since XfXX′ = X ′ we
have the following graph isomorphism from BX onto BX′ .
ϕXX′ : YX → YX′
a −→ a′
b −→ b′
c −→ c′
d −→ d′
So condition (G1) holds, because, {X,X ′} and {X ′′} are the orbits of F on Π. More-
over, there exist the graph epimorphisms πX : BX → Γ(X), πX′ : BX′ → Γ(X ′), and
πX′′ : BX′′ → Γ(X ′′) such that {π−1X (x) | x ∈ X} = {{a, c}, {b, d}}, {π
−1
X′ (x) | x ∈
X ′} = {{a′, c′}, {b′, d′}}, and {π−1X′′(x) | x ∈ X ′′} = {{b′′, c′′}, {a′′, d′′}}. If we de-
fine the epimorphisms ηX : Aut(BX) → FX/F(X), ηX′ : Aut(BX′) → FX
′
/F(X′), and
ηX′′ : Aut(BX′′) → FX
′′
/F(X′′) by{
ηX(idYX ) = F(X)
ηX((ab)(cd)) = (23)F(X){
ηX′(idYX′ ) = F(X′)
ηX′((a
′b′)(c′d′)) = (56)F(X′)
and {
ηX′′(idYX′′ ) = F(X′′)
ηX′′((a
′′b′′)(c′′d′′)) = (14)(25)(36)F(X′′)
304 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
1
2 3
4
6 5 Γ ca
b d
BX c
′a′
b′ d′
BX′ c
′′a′′
b′′ d′′
BX′′
Figure 7: Graph Γ and set of graphs {BX}X∈Π.
then it is easy to verify that condition (G2) holds.
Put Y = {a, b, c, d, a′, b′, c′, d′, a′′, b′′, c′′, d′′}. Consider element g = (2536)(14) ∈ F .
Since X = {2, 3}, X ′ = {5, 6}, and X ′′ = {1, 4} we have Xg = X ′, X ′g = X and
X ′′
g
= X ′′. Now we associate to g, an element g such that
(i) (yX)g = ϕXX′(yX), since ηX(KX) = f−1XX′gF(X) = (56)F(X) = F(X);
(ii) (yX′)g = ϕX′X((yX′)(a
′b′)(c′d′)), since ηX′((a′b′)(c′d′)KX′) = f−1X′XgF(X′) =
(56)F(X′);
(iii) (yX′′)g = (yX′′)(a
′′b′′)(c′′d′′), since ηX′′((a′′b′′)(c′′d′′)KX′′) = gF(X′′) =
(14)(25)(36)F(X′′).
Then g = (aa′bb′)(cc′dd′)(a′′b′′)(c′′d′′). Similarly,
(1) if g = (23) then g = (ab)(cd);
(2) if g = (56) then g = (a′b′)(c′d′);
(3) if g = (23)(56) then g = (ab)(cd)(a′b′)(c′d′);
(4) if g = (2635)(14) then g = (ab′ba′)(cd′dc′)(a′′b′′)(c′′d′′);
(5) if g = (14)(25)(36) then g = (aa′)(bb′)(cc′)(dd′)(a′′b′′)(c′′d′′);
(6) if g = (14)(26)(35) then g = (ab′)(ba′)(cd′)(dc′)(a′′b′′)(c′′d′′).
Since KX , KX′ and KX′′ are trivial groups, it follows that
Aut(Γ) ◦ {Aut(BX)}X∈Π = ⟨idY , (ab)(cd), (a′b′)(c′d′), (aa′bb′)(cc′dd′)(a′′b′′)(c′′d′′),
(ab′ba′)(cd′dc′)(a′′b′′)(c′′d′′), (aa′)(bb′)(cc′)(dd′)(a′′b′′)(c′′d′′),
(ab′)(ba′)(cd′)(dc′)(a′′b′′)(c′′d′′)⟩.
4 Automorphism group of the generalized X-join of graphs
In this section we show that the automorphism group of some graphs which are isomorphic
to a generalized X-join can be expressed in terms of the generalized wreath product of
automorphism groups of its factors.
J. Bagherian: Generalized X-join of graphs and their automorphisms 305
Theorem 4.1. With the notation above, suppose that a graph Γ = (V,R) and a set of
graphs {BX = (YX , EX) | X ∈ Π} satisfy the conditions (G1) and (G2). Then
Aut(Γ) ◦ {Aut(BX)}X∈Π ≤ Aut(Γ ◦π {BX}X∈Π).
Proof. Let W = (Y,E) be the generalized X-join of Γ and {BX}X∈Π with respect to π,
and H = ⟨K,F ⟩ be the generalized wreath product of {Aut(BX)}X∈Π and Aut(Γ).
We show that for every h ∈ H and u, v ∈ Y , if (u, v) ∈ E, then (uh, vh) ∈ E. To do
this, we assume that (u, v) ∈ E and we consider the following cases.
Case 1. Suppose that h =
∏
X∈Π kX ∈ K.
(i) If u, v ∈ YX for someX ∈ Π, then since for everyX ∈ Π, kX ∈ Aut(BX) we have
(u, v)h = (uh, vh) = (ukX , vkX ) ∈ EX .
(ii) If u ∈ YX and v ∈ YX′ for some X and X ′ in Π where X ̸= X ′, then (x, x′) =
(πX(u), πX′(v)) ∈ R and since (ukX , vkX′ ) ∈ π−1X (x)×π
−1
X′ (x
′) we have (u, v)h =
(uh, vh) = (ukX , vkX′ ) ∈ E.
Case 2. Suppose that h = g for some g ∈ F and u, v ∈ YX for some X ∈ Π.
(i) If Xg = X , then since (u, v)g = (ug, vg) = (ut, vt) where ηX(tKX) = gF(X) for
some t ∈ Aut(BX), we have (u, v)h = (uh, vh) = (ut, vt) ∈ E.
(ii) If Xg = X ′ for some X ′ ∈ Π, then since (u, v)g = (ug, vg) = (ϕXX′(ut),
ϕXX′(v
t)) where ηX(tKX) = f−1XX′gF(X) for some t ∈ Aut(BX) we have
(u, v)h = (uh, vh) = (ϕXX′(u
t), ϕXX′(v
t)) ∈ E.
Case 3. Let h = g for some g ∈ F , u ∈ YX and v ∈ YX′ for some X,X ′ ∈ Π where
X ̸= X ′. In this case since (x, x′) = (πX(u), πX′(v)) ∈ R and g ∈ Aut(Γ) we have
(xg, x′
g
) ∈ R. Then the following cases arise.
(i) If Xg = X and X ′g = X ′, then (u, v)g = (ug, vg) = (ut, vt
′
) where ηX(tKX) =
gF(X) and ηX′(t′KX′) = gF(X′). Since πX(ut) = πX(u)ηX(t) = xg and
πX′(v
t′) = πX′(v)
ηX′ (t
′) = x′
g we have
(ut, vt
′
) ∈ π−1X (x
g)× π−1X′ (x
′g).
Then (uh, vh) = (ut, vt
′
) ∈ E.
(ii) If Xg = X and X ′g = X ′′, then (u, v)g = (ug, vg) = (ut, ϕX′X′′(vt
′
)) where
ηX(tKX) = gF(X) and ηX′(t′KX′) = f
−1
X′X′′gF(X′). Then πX(u
t) =
πX(u)
ηX(t) = xg and by condition (G2) we have
πX′′(ϕX′X′′(v
t′)) = ξX′X′′(πX′(v
t′))
= ξX′X′′(πX′(v)
ηX′ (t
′))
= fX′X′′ηX′(t
′KX′)(πX′(v))
= gF(X′)(πX′(v))
= (πX′(v))
g
= x′
g
.
306 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
So (ut, ϕX′X′′(vt
′
)) ∈ π−1X (xg)×π
−1
X′′(x
′g) and then (uh, vh) = (ut, ϕX′X′′(vt
′
)) ∈
E.
(iii) If Xg = X ′ and X ′g = X ′′, then (u, v)g = (ug, vg) = (ϕXX′(ut), ϕX′X′′(vt
′
))
where ηX(tKX) = f−1XX′gF(X) and ηX′(t
′KX′) = f
−1
X′X′′gF(X′). From condition
(G2) we have
πX′(ϕXX′(u
t)) = ξXX′(πX(u
t))
= ξXX′(πX(u)
ηX(t))
= fXX′ηX(tKX)(πX(u))
= gF(X)(πX(u))
= (πX(u))
g
= xg.
Similarly, πX′′(ϕX′X′′(vt
′
)) = x′
g
. Then
(ϕXX′(u
t), ϕX′X′′(v
t′)) ∈ π−1X′ (x
g)× π−1X′′(x
′g)
and so (uh, vh) = (ϕXX′(ut), ϕX′X′′(vt
′
)) ∈ E.
(iv) If Xg = X ′′ and X ′g = X ′′′, then (u, v)g = (ug, vg) = (ϕXX′′(ut), ϕX′X′′′(vt
′
))
where ηX(tKX) = f−1XX′′gF(X) and ηX′(t
′KX′) = f
−1
X′X′′′gF(X′). Then an argu-
ment similar to that given in (iii) shows that πX′′(ϕXX′′(ut)) = xg and
πX′′′(ϕX′X′′′(v
t′)) = x′
g . So (ϕXX′′(ut), ϕX′X′′′(vt
′
)) ∈ π−1X′′(xg) × π
−1
X′′′(x
′g)
and hence (uh, vh) = (ϕXX′′(ut), ϕX′X′′′(vt
′
)) ∈ E.
Then we conclude that ⟨K,F ⟩ ⊆ Aut(Γ ◦π {BX}X∈Π). Thus
Aut(Γ) ◦ {Aut(BX)}X∈Π ≤ Aut(Γ ◦π {BX}X∈Π).
The inclusion Aut(Γ)◦{Aut(BX)}X∈Π ≤ Aut(Γ◦π{BX}X∈Π) in the above theorem
may be proper. For example
D8 = Aut(K2) ◦Aut(K2) < Aut(K2 ◦K2) = S4,
where S4 is the symmetric group on 4 elements V = {1, 2, 3, 4} and D8 = {idV , (12),
(34), (12)(34), (13)(24), (14)(23), (1324), (1423)} is the dihedral group of order 8; see
[6, Chapter 10]. In the following we give necessary and sufficient conditions under which
the above inclusion is proper.
Theorem 4.2. With the notation above, suppose that the graph Γ = (V,R) and the set
of graphs {BX = (YX , EX) | X ∈ Π} satisfy the conditions (G1) and (G2). Let W =
(Y,E) be the generalized X-join of Γ and {BX}X∈Π with respect to π and let E0 ⊆ E1 be
the equivalence relations defined in (1) and (2). Then the inclusion
Aut(Γ) ◦ {Aut(BX)}X∈Π ≤ Aut(W )
is proper if and only if there exist equivalence relations E′0 ⊆ E′1 on Y such that the
following conditions hold.
J. Bagherian: Generalized X-join of graphs and their automorphisms 307
(i) E′0 ⊊ E0 and E′1 ⊊ E1, and W = Γ′ ◦π′ {AZ}Z∈Π′ , where the graph Γ′ is the
quotient graph W/E′0, Π
′ is a partition of V (Γ′), {AZ}Z∈Π′ are the subgraphs of
W induced by the equivalence classes of E′1, π
′ =
⋃̇
Z∈Π′π
′
Z , and {π′
−1
Z (x) | x ∈
V (Γ′), Z ∈ Π′} is the set of equivalence classes of E′0.
(ii) There exist Z ̸= Z ′ ∈ Π′−Π and a graph isomorphism ϕ : YZ → YZ′ fromAZ onto
AZ′ , such that ϕ preserves equivalence classes of E′0 contained in YZ .
(iii) For every equivalence class S′ of E′0 contained in YZ , if W [S
′] is a union of con-
nected components of W [S] for some S ∈ W/E0, where S′ ⊊ S then π′Z(S′) and
π′Z′(ϕ(S
′)) are nonadjacent, otherwise π′Z(S
′) and π′Z′(ϕ(S
′)) are adjacent. In
both cases π′Z(S
′) and π′Z′(ϕ(S
′)) have the same neighbors in V (Γ′) \ Z ∪ Z ′.
(iv) For each two distinct equivalence classes S′1 , S
′
2 of E
′
0 that are contained in YZ , S
′
1
and ϕ(S′2) are adjacent if and only if ϕ(S
′
1) and S
′
2 are adjacent.
Proof. Suppose that there exists ϕ ∈ Aut(W ) \ Aut(Γ) ◦ {Aut(BX)}X∈Π. Let U be
the set of all elements of Y that are moved by ϕ. Since Y/E1 = {YX | X ∈ Π} and
Y/E0 =
⋃
X∈Π ΛX , where ΛX = {π
−1
X (x) | x ∈ X}, are two system of blocks for
Aut(Γ) ◦ {Aut(BX)}X∈Π, then there exist
(1) X ∈ Π such that UX = U ∩ YX ̸= ∅;
(2) X ′ ∈ Π such that X ̸= X ′ and ϕ(UX) ⊆ YX′ . Note that if X = X ′, then the
restriction of ϕ to YX is an automorphism of BX and ϕ(UX) ⊆ YX . But since
ϕ /∈ Aut(Γ) ◦ {Aut(BX)}X∈Π we must have at least two equivalence classes
S1, S2 ∈ E0 such that a part of S1 is moved by the automorphism ϕ to a part of
S2. This contradicts the fact that ΛX = {π−1X (x) | x ∈ X} is a system of blocks for
Aut(BX).
(3) at least one equivalence class S of E0 such that S ∩ UX ⊊ S. Indeed, if UX =⋃t
i=1 Si ⊊ YX where every Si is an equivalence class of E0, then by (2), ϕ(UX) ⊆
YX′ for some X ′ ̸= X and ϕ(UX) is a union of some equivalence classes of E0
which are contained in YX′ . This means that the vertices πX(S1), . . . , πX(St) of X
can be moved to the vertices πX′(ϕ(S1)), . . . , πX′(ϕ(St)) of X ′. This contradicts
the fact that Π is a system of blocks for Aut(Γ).
Put VX′ = ϕ(UX). Let S1, S2, ..., St be the equivalence classes of E0 contained in YX
such that for every 1 ≤ i ≤ t,
SXi = Si ∩ UX ̸= ∅.
Then for at least one i, SXi ⊊ Si. Moreover, we have the following.
(a) The restriction of ϕ to UX gives an isomorphism between W [UX ] and W [VX′ ], the
subgraphs of W induced by UX and VX′ .
(b) For each i the vertices in SXi ∪ ϕ(SXi ) have the same neighbors in Y \ (UX ∪ VX′).
Indeed, suppose that u ∈ SXi and w is a neighbor of u. Suppose that T1, . . . , Tt
are equivalence classes of E0 such that ϕ(SXi ) ∩ Ti ̸= ∅ and vi ∈ Ti \ ϕ(SXi ).
If w ∈ Y \ (YX ∪ YX′), then ϕ(w) is adjacent to all vertices of Ti, specially vi.
So w and ϕ−1(vi) = vi are adjacent. Thus w is adjacent to all vertices of ϕ(SXi ).
308 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
Moreover, if w ∈ YX \ UX , then since w is adjacent to u, ϕ(w) = w is adjacent to
ϕ(u). Then w is adjacent to all vertices of ϕ(SXi ). Similarly, if w ∈ YX′ \ VX′ , then
w is adjacent to all vertices of ϕ(SXi ). Hence we conclude that S
X
i ∪ ϕ(SXi ) have
the same neighbors in Y \ (UX ∪ VX′).
(c) If W [SXi ] is a union of connected components of W [Si], then S
X
i and ϕ(S
X
i ) are
nonadjacent; otherwise by the definition of W , Si \ SXi and ϕ(SXi ) are adjacent and
since ϕ ∈ Aut(W ), SXi and Si \SXi are adjacent and this contradicts the hypothesis
that W [SXi ] is a union of connected components of W [Si]. Also if W [S
X
i ] and
W [Si \ SXi ] are adjacent, then since ϕ ∈ Aut(W ), Si \ SXi and ϕ(SXi ) must be
adjacent and the definition of W implies that ϕ(SXi ) and Si are externally related to
each other. Moreover, Si \ SXi and SXi are also externally related to each other.
(d) For two different equivalence classes S1 and S2 of E0 with SX1 , S
X
2 ̸= ∅, if S1 and
ϕ(SX2 ) are adjacent then from the definition of W it follows that S1 and ϕ(S
X
2 ) are
externally related to each other. Moreover, since ϕ ∈ Aut(W ) we must have ϕ(SX1 )
and S2 are also externally related to each other. Similarly, if S2 and ϕ(SX1 ) are
adjacent then ϕ(SX2 ) and S1 are externally related to each other.
Now we consider two equivalence relations E′0 ⊆ E′1 on Y such that
Y/E′1 = {UX , VX′ , YX \ UX , YX′ \ VX′ , Y/E1 \ {YX , YX′}},
and the equivalence classes of E′0 are equal
SXi , Si \ SXi , ϕ(SXi ), Ti \ ϕ(SXi ), 1 ≤ i ≤ t,
and Y/E0 \ {Si, Ti | 1 ≤ i ≤ t}, where for every i, Ti is an equivalence class of E0 such
that ϕ(SXi ) ⊆ Ti.
From statements (b) and (c) we conclude that the condition (ii) of Theorem 2.4 holds and so
W = Γ′ ◦π′ {AX}X∈Π′ , where Γ′ is the quotient graph W/E′0, Π′ is a partition of V (Γ′)
induced by Y/E′1, and {AX}X∈Π′ are the subgraphs of W induced by the equivalence
classes of E′1. So (i) holds. If we denote by AX and AX′ the subgraphs of W induced
by UX and VX′ , respectively, then the restriction of ϕ to UX gives a graph isomorphism
betweenAX andAX′ . Clearly, ϕ preserves the equivalence classes of E′0 contained in UX .
Thus condition (ii) of theorem holds. Moreover, (b), (c) and the definition of π′ imply that
condition (iii) holds. Finally, condition (iv) follows from statement (d).
Conversely, suppose that there exist equivalence relations E′0 ⊆ E′1 on Y such that
conditions (i) – (iv) hold. Let UZ and UZ′ be the vertex sets of AZ and AZ′ , respectively.
Assume that ϕ : UZ −→ UZ′ is the graph isomorphism from AZ onto AZ′ . We define a
bijection ψ : Y −→ Y as follows:
ψ(v) =

ϕ(v) if v ∈ UZ ,
ϕ−1(v) if v ∈ UZ′ ,
v if v /∈ {UZ , UZ′}.
We claim that ψ ∈ Aut(W ). To do this, we suppose that (u, v) ∈ E and we consider the
following cases.
J. Bagherian: Generalized X-join of graphs and their automorphisms 309
(1) If u, v ∈ Y \ UZ ∪ UZ′ , then clearly (ψ(u), ψ(v)) = (u, v) ∈ E.
(2) If u, v ∈ UZ , then since ϕ is a graph isomorphism it follows that (ψ(u), ψ(v)) =
(ϕ(u), ϕ(v)) ∈ E. Similarly, if u, v ∈ UZ′ we have (ψ(u), ψ(v)) = (ϕ−1(u),
ϕ−1(v)) ∈ E.
(3) If u ∈ UZ and v ∈ Y \ UZ ∪ UZ′ , then since v is a neighbor of u it follows from
(iii) that v is also a neighbor of ϕ(u). Hence (ψ(u), ψ(v)) = (ϕ(u), v) ∈ E.
(4) If u ∈ UZ and v ∈ UZ′ such that for some equivalence class S′ of E′0, u ∈ S′ and
v ∈ ϕ(S′), then the definition of W implies that all vertices in S′ are adjacent to all
vertices in ϕ(S′) and so (ψ(u), ψ(v)) = (ϕ(u), ϕ−1(v)) ∈ E.
(5) If u ∈ UZ and v ∈ UZ′ such that for two equivalence classes S′1 and S′2 of E′0,
u ∈ S′1 and v ∈ ϕ(S′2), then by the definition of W , all vertices in S′1 are adjacent
to all vertices in ϕ(S′2). On the other hand, it follows from (iv) that ϕ(S
′
1) and S
′
2
are adjacent. So all vertices of ϕ(S′1) are adjacent to all vertices of S
′
2 and thus
(ψ(u), ψ(v)) = (ϕ(u), ϕ−1(v)) ∈ E.
Hence ψ ∈ Aut(W ). SinceE′0 ⊊ E0 and Z,Z ′ ∈ Π′−Π, and ψ preserves the equivalence
classes of E′0 we conclude that ψ ∈ Aut(W ) \Aut(Γ) ◦ {Aut(BX)}X∈Π.
Example 4.3. Suppose that Γ is the graph in Figure 8 with vertices V = {1, 2, 3, 4, 5, 6}.
Then one can see that
F = Aut(Γ) = {idV , (12)(56)(34), (13)(24), (23)(56)(14)}
and
Π = {X = {1, 2}, X ′ = {3, 4}, X ′′ = {5, 6}},
is a system of blocks for F . Moreover, FX = FX
′
= {idV , (12)(56)(34)}, FX
′′
= F ,
F(X) = F(X′) = {idV }, and F(X′′) = {idV , (13)(24)}. Suppose that BX , BX′ , and
BX′′ are the graphs in Figure 8 with vertices YX = {a, b, c, d}, YX′ = {a′, b′, c′, d′},
YX′′ = {a′′, b′′, c′′, d′′}, respectively.
Now consider the graph epimorphisms πX : BX → Γ(X), πX′ : BX′ → Γ(X ′), and
πX′′ : BX′′ → Γ(X ′′) as the following:
1
3 4
2
5 6
Γ
a
bc
d BX
d′
c′ b′
a′
BX′ a
′′ b′′
c′′ d′′
BX′′
Figure 8: Graph Γ and set of graphs {BX}X∈Π.
{
πX(a) = πX(b) = 1
πX(c) = πX(d) = 2
310 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
{
πX′(a
′) = πX′(b
′) = 3
πX′(c
′) = πX′(d
′) = 4
and {
πX′′(a
′′) = πX′′(b
′′) = 6
πX′′(c
′′) = πX′′(d
′′) = 5
.
Since Aut(BX) = {idYX , (bc)(ad)}, Aut(BX′) = {idYX′ , (b
′c′)(a′d′)}, and
Aut(BX′′) = {idYX′′ , (a
′′b′′), (c′′d′′), (a′′b′′)(c′′d′′), (a′′c′′b′′d′′), (a′′d′′b′′c′′)
, (a′′c′′)(b′′d′′), (a′′d′′)(b′′c′′)},
we can define epimorphisms ηX : Aut(BX) → FX/F(X), ηX′ : Aut(BX′) → FX
′
/F(X′),
and ηX′′ : Aut(BX′′) → FX
′′
/F(X′′) by{
ηX(idYX ) = idV
ηX((bc)(ad)) = (12)(56)(34)F(X){
ηX′(idYX′ ) = idV
ηX′((b
′c′)(a′d′)) = (12)(56)(34)F(X′)
and
ηX′′(idYX′′ ) = ηX′′((a
′′b′′)) = ηX′′((c
′′d′′)) = ηX′′((a
′′b′′)(c′′d′′)) = idV
ηX′′((a
′′c′′b′′d′′)) = ηX′′((a
′′d′′b′′c′′)) = ηX′′((a
′′c′′)(b′′d′′))= ηX′′((a
′′d′′)(b′′c′′))=
(12)(56)(34)F(X′′)
Then KX and KX′ are trivial groups and
KX′′ = {idYX′′ , (a
′′b′′), (c′′d′′), (a′′b′′)(c′′d′′)}.
Let TX = {idYX , (bc)(ad)}, TX′ = {idYX′ , (b
′c′)(a′d′)}, and TX′′ = {idYX′′ ,
(a′′c′′)(b′′d′′)}. Put fXX′ = (13)(24). Then the elements (12)(56)(34), (13)(24) and
(23)(56)(14) in Aut(Γ) are associated to (bc)(ad)(b′c′)(a′d′)(a′′c′′)(b′′d′′),
(aa′)(bb′)(cc′)(dd′) and (bc′)(ad′)(cb′)(da′)(a′′c′′)(b′′d′′), respectively. Then we have
Aut(Γ)◦{Aut(BX)}X∈Π=⟨idY , (aa′)(bb′)(cc′)(dd′), (bc′)(ad′)(cb′)(da′)(a′′c′′)(b′′d′′),
(a′′b′′), (c′′d′′), (bc)(ad)(b′c′)(a′d′)(a′′c′′)(b′′d′′)⟩.
Now letW = (Y,E) be the generalized X-join of Γ and {BX}X∈Π with respect to π. (See
Figure 9.) Consider the equivalence relations E′0 and E
′
1 on Y with the following classes,
Y/E′0 = {{a}, {b}, {c}, {d}, {a′}, {b′}, {c′}, {d′}, {a′′, b′′}, {c′′, d′′}}
Y/E′1 = {{a, d}, {b, c}, {a′, d′}, {b′, c′}, {a′′, b′′, c′′, d′′}}.
Then one can see that
J. Bagherian: Generalized X-join of graphs and their automorphisms 311
a
b
c
d
a′′ b′′
d′
c′
b′
a′
c′′ d′′
Figure 9: Graph W = Γ ◦π {BX}X∈Π.
(1) Γ ◦π {BX}X∈Π = Γ′ ◦π′ {AZ}Z∈Π′ , where Γ′ is the quotient graph W/E′0 with
vertices V (Γ′) = {1, 2, . . . , 10}, and {AZ}Z∈Π′ are the subgraphs of W induced by
the equivalence classes of E′1, and π
′ =
⋃̇
Z∈Π′π
′
Z maps {a}, {b}, {c}, {d}, {a′},
{b′}, {c′}, {d′}, {c′′, d′′}, {a′′, b′′} onto 1, 2, . . . , 10, respectively. (See Figure 10.)
(2) Put YZ = {b, c} and YZ′ = {b′, c′} and let AZ =W [YZ ] and AZ′ =W [YZ′ ]. Then
ϕ : YZ → YZ′ such that ϕ(b) = b′ and ϕ(c) = c′ is a graph isomorphism from AZ
onto AZ′ . Clearly, ϕ preserves the equivalence classes of E′0 contained in YZ .
(3) YZ contains two equivalence classes S′1 = {b} and S′2 = {c} of E′0 such that S′1 ⊆
S1 and S′2 ⊆ S2 where S1 = {a, b} ∈ Y/E0 and S2 = {c, d} ∈ Y/E0. Moreover,
W [S1] is connected and π′Z(b) and π
′
Z′(ϕ(b)) are adjacent and have the same neigh-
bors in V (Γ′) \ {Z ∪Z ′}, where Z = {π′Z(b), π′Z(c)} and Z ′ = {π′Z′(b′), π′Z′(c′)}.
Similarly, W [S2] is connected and π′Z(c) and π
′
Z′(ϕ(c)) are adjacent and have the
same neighbors in V (Γ′) \ {Z ∪ Z ′}.
(4) The vertex b is nonadjacent to ϕ(c) and vertex c is nonadjacent to ϕ(b).
Then the conditions of Theorem 4.2 hold. So
Aut(Γ) ◦ {Aut(BX)}X∈Π ⪇ Aut(Γ ◦π {BX}X∈Π).
In the following as a main result, we give necessary and sufficient conditions under
which the full automorphism group of the generalized X-join of graphs is equal to the
generalized wreath product of the automorphism groups of their factors.
312 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
9
1
2
3
4
10
8
7
6
5
b
c
a
d
b′
c′
a′
d′
a′′
b′′
c′′
d′′
Figure 10: Graph Γ′ and set of graphs {AZ}Z∈Π′ .
Corollary 4.4. Suppose that W = Γ ◦π {BX}X∈Π is such that the graph Γ = (V,R) and
the set of graphs {BX = (YX , EX) | X ∈ Π} satisfy the conditions (G1) and (G2). Then
Aut(Γ◦π {BX}X∈Π) = Aut(Γ)◦{Aut(BX)}X∈Π if and only if there are no equivalence
relations E′0 ⊆ E′1 on Y satisfying the conditions (i), (ii), (iii), and (iv) of Theorem 4.2.
Proof. This follows immediately from Theorem 4.2.
Example 4.5. Let W = (Y,E) be the graph in Figure 11. It is easy to see that W is the
graph Γ ◦π {BX , BX′ , BX′′} where Γ and {BX , BX′ , BX′′}, and π = πX ∪ πX′ ∪ πX′′
are given in Example 3.3. Moreover, Y/E0 = {{a, c}, {b, d}, {a′, c′}, {b′, d′}, {b′′, c′′},
{a′′, d′′}} and Y/E1 = {{a, c, b, d}, {a′, c′, b′, d′}, {b′′, c′′, a′′, d′′}}. Since there are no
equivalence relations E′0 ⊆ E′1 on Y that satisfy the conditions (i), (ii), (iii), and (iv) of
Theorem 4.2, it follows that
Aut(W ) = Aut(Γ)◦{Aut(BX)}X∈Π = ⟨idY , (ab)(cd), (aa′bb′)(cc′dd′)(a′′b′′)(c′′d′′),
(a′b′)(c′d′), (ab′ba′)(cd′dc′)(a′′b′′)(c′′d′′),
(aa′)(bb′)(cc′)(dd′)(a′′b′′)(c′′d′′),
(ab′)(ba′)(cd′)(dc′)(a′′b′′)(c′′d′′)⟩.
The next corollary follows directly from Theorem 4.2.
Corollary 4.6. Suppose that the graph Γ = (V,R) and the set of graphs {BX = (YX , EX) |
X ∈ Π} satisfy the conditions (G1) and (G2). Let W = (Y,E) be the generalized X-join
of Γ and {BX}X∈Π with respect to π and letE0 ⊆ E1 be the equivalence relations defined
in (1) and (2). Then
Aut(Γ) ◦ {Aut(BX)}X∈Π = Aut(Γ ◦π {BX}X∈Π)
if W is uniquely determined by E0 and E1.
Corollary 4.7 (See [8, Theorem 2.10]). Let Γ = (V,R) be a graph and {Bx | x ∈ V } be
a set of graphs such that Bx ≃ Bx′ whenever xf = x′ for some f ∈ Aut(Γ). Then
Aut(Γ[Bx]x∈V ) = Aut(Γ) ◦ {Aut(Bx)}x∈V
if and only if
J. Bagherian: Generalized X-join of graphs and their automorphisms 313
d′′
d b a c
a′′
b′′
c′′
d′ b′ a′ c′
Figure 11: Graph W .
(1) Bx is connected if there exists at least one vertex w ∈ V such that x and w are
nonadjacent and have the same neighbors in V ,
(2) Bx is connected if there exists at least one vertex w ∈ V such that x and w are
adjacent and have the same neighbors in V \ {x,w}.
Proof. By Example 2.2, W = Γ[Bx]x∈V = Γ ◦π {Bx}x∈V where Π = {{x} | x ∈ V }
and πx : Yx → X is a graph epimorphism from Bx onto Γ(X) such that πx(yx) = x for
every yx ∈ Yx. In this case E0 = E1 and FX = F(X). If we define
ηX := Aut(BX) → FX/F(X)
by ηX(α) = 1FX/F(X) for every α ∈ Aut(BX), then ηX is an epimorphism and condition
(G2) holds. Then it follows from Corollary 4.4 that
Aut(Γ[Bx]x∈V ) = Aut(Γ) ◦ {Aut(Bx)}x∈V
if and only if there is no equivalence relation E′0 on Y satisfying the conditions (i), (ii),
(iii), and (iv) of Theorem 4.2.
Now suppose that Aut(Γ[Bx]x∈V ) = Aut(Γ) ◦ {Aut(Bx)}x∈V and there exist x,w ∈
V such that x and w are nonadjacent and have the same neighbors in V . If Bx is discon-
nected then Bw is also disconnected and we can define an equivalence relation E′0 on Y
such that the equivalence classes of E′0 are Yz, z ̸∈ {x,w}, together with the connected
components of Bx and Bw. Since x and w are nonadjacent and have the same neighbors in
V , one can see that the conditions (i), (ii), (iii), and (iv) of Theorem 4.2 hold, a contradic-
tion. So Bx is connected. Moreover, suppose that there exist x,w ∈ V such that x and w
are adjacent and have the same neighbors in V \ {x,w}. If Bx is disconnected, then there
exist at least two subsets S1, S2 ⊂ Yx such that all vertices of S1 are adjacent to all vertices
of S2. Similarly, there exist subsets S′1, S
′
2 ⊂ Yw with the property that all vertices of S′1
are adjacent to all vertices of S′2. Then we can define an equivalence relation E
′
0 on Y such
that S1, S2, S′1, and S
′
2 together with Yz, z ̸∈ {x,w} are its equivalence classes. One can
see that in this case the conditions (i), (ii), (iii), and (iv) of Theorem 4.2 hold and thus again
we have a contradiction.
314 Ars Math. Contemp. 24 (2024) #P2.06 / 293–315
Conversely, suppose that conditions (1) and (2) hold and suppose on the contrary that
φ ∈ Aut(Γ[Bx]x∈V ) \Aut(Γ) ◦ {Aut(Bx)}x∈V .
Then there is an equivalence relation E′0 on Y satisfying the conditions (i), (ii), (iii), and
(iv) of Theorem 4.2. It follows from condition (i) that W = Γ′ ◦π′ {Az}z∈V (Γ′), where
the graph Γ′ is the quotient graphW/E′0 and {Az}z∈V (Γ′) are the subgraphs ofW induced
by the equivalence classes of E′0. It follows from (ii) that there exist x,w ∈ V such that
the equivalence classes of E′0 contain Yz ⊊ Yx and Yz′ = φ(Yz) ⊊ Yw. By (iii) if Bx[Yz]
is a union of connected components of Bx, then z and z′ are nonadjacent and all of their
neighbors are exactly the same in V (Γ′), otherwise z and z′ are adjacent and have the
same neighbors in V (Γ′) \ {z, z′}. This implies that if Bx is disconnected then z and z′
are nonadjacent and all vertices in Yz and all vertices in Yz′ have the same neighbors in
Y \ (Yz ∪ Yz′). Since Yz ⊊ Yx and Yz′ ⊊ Yw it follows that x and w must be nonadjacent
and have the same neighbors in V , which contradicts (1). Moreover, if Bx is connected,
since z and z′ are adjacent and have the same neighbors in V (Γ′) \ {z, z′} it follows that
all vertices in Yz are adjacent to all vertices of Yz′ and all vertices in Yz and all vertices in
Yz′ have the same neighbors in Y \ (Yz ∪ Yz′). Then all vertices in Yx are adjacent to all
vertices of Yw. So x and w must be adjacent and have the same neighbors in V \ {x,w}.
Furthermore, since all vertices in Yz are adjacent to all vertices of Yx \ Yz it follows that
Bx is disconnected, which contradicts (2). Thus we have
Aut(Γ[Bx]x∈V ) = Aut(Γ) ◦ {Aut(Bx)}x∈V .
5 Conclusion
A generalization of the X-join of graphs has been introduced and necessary and sufficient
conditions under which a graph is isomorphic to a generalized X-join has been given. A
generating set for the automorphism groups of a class of graphs which are isomorphic to a
generalized X-join has been computed.
Since the generalized X-join of graphs is a natural generalization of the X-join of
graphs, the results on the X-join or lexicographic product of graphs can be also studied
for the generalized X-join of graphs.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.07 / 317–325
https://doi.org/10.26493/1855-3974.2126.5b3
(Also available at http://amc-journal.eu)
The automorphism group of the zero-divisor
digraph of matrices over an antiring
David Dolžan *
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska 21, 1000 Ljubljana, Slovenia and
IMFM, Jadranska 19, 1000 Ljubljana, Slovenia
Gabriel Verret
Department of Mathematics, University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand
Received 26 September 2019, accepted 29 May 2023, published online 3 October 2023
Abstract
We determine the automorphism group of the zero-divisor digraph of the semiring of
matrices over an antinegative commutative semiring with a finite number of zero-divisors.
Keywords: Automorphism group of a graph, zero-divisor graph, semiring.
Math. Subj. Class. (2020): 05C60, 16Y60, 05C25
1 Introduction
In recent years, the zero-divisor graphs of various algebraic structures have received a lot
of attention, since they are a useful tool for revealing the algebraic properties through their
graph-theoretical properties. In 1988, Beck [3] first introduced the concept of the zero-
divisor graph of a commutative ring. In 1999, Anderson and Livingston [1] made a slightly
different definition of the zero-divisor graph in order to be able to investigate the zero-
divisor structure of commutative rings. In 2002, Redmond [15] extended this definition to
also include non-commutative rings. Different authors then further extended this concept
to semigroups [6], nearrings [4] and semirings [8].
Automorphisms of graphs play an important role both in graph theory and in algebra,
and finding the automorphism group of certain graphs is often very difficult. Recently, a
*Corresponding author. The author acknowledges the financial support from the Slovenian Research Agency
(research core funding no. P1-0222).
E-mail addresses: david.dolzan@fmf.uni-lj.si (David Dolžan), g.verret@auckland.ac.nz (Gabriel Verret)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
318 Ars Math. Contemp. 24 (2024) #P2.07 / 317–325
lot of effort has been made to determine the automorphism group of various zero-divisor
graphs. In [1], Anderson and Livingston proved that Aut(Γ(Zn)) is a direct product of
symmetric groups for n ≥ 4 a non-prime integer. In the non-commutative case, the case of
matrix rings and semirings is especially interesting. Thus, it was shown in [10] that, when p
is a prime, Aut(Γ(M2(Zp))) is isomorphic to Sym(p+ 1), the symmetric group of degree
p+1. More generally, it was proved in [13], that Aut(Γ(M2(Fq))) ∼= Sym(q+1). In [18],
the authors determined the automorphism group of the zero-divisor graph of all rank one
upper triangular matrices over a finite field, and in [16] they determined the automorphism
group of the zero-divisor graph of the matrix ring of all upper triangular matrices over a
finite field. Recently, the automorphism group of the zero-divisor graph of the complete
matrix ring of matrices over a finite field has been found independently in [17] and [20].
In this paper, we study the zero-divisor graph of matrices over commutative semirings.
The theory of semirings has many applications in optimization theory, automatic control,
models of discrete event networks and graph theory (see e.g. [2, 5, 12, 19]) and the zero-
divisor graphs of semirings were recently studied in [9, 7, 14]. For an extensive theory of
semirings, we refer the reader to [11]. There are many natural examples of commutative
semirings, for example, the set of nonnegative integers (or reals) with the usual operations
of addition and multiplication. Other examples include distributive lattices, tropical semir-
ings, dioı̈ds, fuzzy algebras, inclines and bottleneck algebras.
The theory of matrices over semirings differs quite substantially from the one over
rings, so the methods we use are necessarily distinct from those used in the ring setting.
The main result of this paper is the determination of the automorphism group of the zero-
divisor digraph of a semiring of matrices over an antinegative commutative semiring with
a finite number of zero-divisors (see Theorem 3.12).
2 Definitions and preliminaries
2.1 Digraphs
A digraph Γ consists of a set V(Γ) of vertices, together with a binary relation → on V(Γ).
An automorphism σ of Γ is a permutation of V(Γ) such that u → v ⇐⇒ σ(u) → σ(v).
The automorphisms of Γ form its automorphism group Aut(Γ).
Let Γ be a digraph and let v ∈ V(Γ). We write N−(v) = {u ∈ V(Γ): u → v} and
N+(v) = {u ∈ V(Γ): v → u}. If, for u, v ∈ V(Γ), we have N−(u) = N−(v) and
N+(u) = N+(v), then we say u and v are twin vertices. The relation ∼ on V(Γ), defined
by u ∼ v if and only if u and v are twin vertices, is clearly an equivalence relation preserved
by Aut(Γ). For v ∈ V(Γ), we shall denote by v the ∼-equivalence class of v. Let Γ be
the graph with these equivalence classes as vertices and u →Γ v if and only if u →Γ v.
For σ ∈ Aut(Γ) we denote by σ the induced automorphism of Γ. An automorphism
σ ∈ Aut(Γ) is called regular if σ is the identity map.
2.2 Semirings
A semiring is a set S equipped with binary operations + and · such that (S,+) is a commu-
tative monoid with identity element 0, and (S, ·) is a semigroup. Moreover, the operations
+ and · are connected by distributivity and 0 annihilates S.
A semiring S is commutative if ab = ba for all a, b ∈ S, and antinegative if, for all
a, b ∈ S, a + b = 0 implies that a = 0 or b = 0. Antinegative semirings are also called
D. Dolžan et al.: The automorphism group of the zero-divisor digraph of matrices . . . 319
zerosum-free semirings or antirings. The smallest nontrivial example of an antiring is the
Boolean antiring B = {0, 1} with addition and multiplication defined so that 1 + 1 =
1 · 1 = 1.
Let S be a semiring. For x ∈ S, we define the left and right annihilators in S by
AnnL(x) = {y ∈ S : yx = 0} and AnnR(x) = {y ∈ S : xy = 0}. If S is commutative,
we simply write Ann(x) for AnnL(x) = AnnR(x). We denote by Z(S) the set of zero-
divisors of S, that is Z(S) = {x ∈ S : ∃y ∈ S \ {0} such that xy = 0 or yx = 0}. The
zero-divisor digraph Γ(S) of S is the digraph with vertex-set S and u → v if and only if
uv = 0.
It is easy to see that if n ≥ 1 and S is a semiring, then the set Mn(S) of n×n matrices
forms a semiring with respect to matrix addition and multiplication. If S is antinegative,
then so is Mn(S). If S has an identity 1, let Eij ∈ Mn(S) with entry 1 in position (i, j),
and 0 elsewhere. For s ∈ S, define sEij ∈ Mn(S) as the matrix with entry s in position
(i, j), and 0 elsewhere.
3 The automorphisms of the zero-divisor digraph
The following fact will be used repeatedly.
Lemma 3.1. Let S be a semiring. If A,B ∈ S and σ ∈ Aut(Γ(S)), then
σ(AnnL(A)) = AnnL(σ(A)) and σ(AnnR(A)) = AnnR(σ(A)).
Proof. We have
X ∈ σ(AnnL(A)) ⇐⇒ σ−1(X) ∈ AnnL(A)
⇐⇒ σ−1(X)A = 0
⇐⇒ Xσ(A) = 0
⇐⇒ X ∈ AnnL(σ(A)).
The proof of the second part is analogous.
Lemma 3.2. Let S be an antiring and let Γ = Γ(S). If A,B ∈ S and σ ∈ Aut(Γ),
then σ(A + B) and σ(A) + σ(B) are twin vertices and, in particular, σ(A+B) =
σ(A) + σ(B).
Proof. Using antinegativity, we have
X ∈ AnnL(σ(A+B)) ⇐⇒ Xσ(A+B) = 0
⇐⇒ σ−1(X)(A+B) = 0
⇐⇒ σ−1(X)A = σ−1(X)B = 0
⇐⇒ Xσ(A) = Xσ(B) = 0
⇐⇒ X(σ(A) + σ(B)) = 0
⇐⇒ X ∈ AnnL(σ(A) + σ(B)).
We have proved that AnnL(σ(A+B)) = AnnL(σ(A)+σ(B)). An analogous proof yields
AnnR(σ(A+B)) = AnnR(σ(A)+σ(B)). This implies that σ(A+B) and σ(A)+σ(B)
are twin vertices.
320 Ars Math. Contemp. 24 (2024) #P2.07 / 317–325
Definition 3.3. Let S be a commutative semiring, let n ∈ N and let A ∈ Mn(S) with
(i, j) entry aij . For every i, j ∈ {1, . . . , n}, we define Ci(A) =
⋂n
k=1 Ann(aki) and
Rj(A) =
⋂n
k=1 Ann(ajk). Let AR(A) := (C1(A), . . . , Cn(A)) ∈ P(S)n and AL(A) :=
(R1(A), . . . , Rn(A)) ∈ P(S)n, where P(S) denotes the power set of S.
The next theorem characterizes the twin vertices of Γ(Mn(S)).
Theorem 3.4. Let S be a commutative antiring, let n ∈ N and let A,B ∈ Mn(S). Then
A and B are twin vertices of Γ(Mn(S)) if and only if AL(A) = AL(B) and AR(A) =
AR(B).
Proof. Let aij and bij be the (i, j) entry of A and B, respectively. Suppose first that A
and B are twin vertices of Γ(Mn(S)) and assume that AR(A) ̸= AR(B). This implies
that, for some i ∈ {1, . . . , n}, we have Ci(A) ̸= Ci(B). Swapping the role of A and B if
necessary, there exists s ∈ S such that s ∈ Ci(A) and s /∈ Ci(B). Therefore, there exists
k ∈ {1, . . . , n} such that s /∈ Ann(bki). Now, let C = sEik ∈ Mn(S) and observe that
AC = 0 but BC ̸= 0, so N+(A) ̸= N+(B), which is a contradiction with the fact that A
and B are twin vertices. We have thus proved that AR(A) = AR(B). A similar argument
yields that AL(A) = AL(B).
Conversely, assume now that AL(A) = AL(B) and AR(A) = AR(B). Suppose there
exists X ∈ Mn(S) such that AX = 0. Therefore, for all i, j ∈ {1, . . . , n} we have∑n
k=1 aikxkj = 0. Since S is an antiring, this further implies that aikxkj = 0 for all
i, j, k ∈ {1, . . . , n}. So, xkj ∈ Ann(aik) and therefore xkj ∈ Ck(A) = Ck(B) for
all k ∈ {1, . . . , n}. Thus, for all i, j, k ∈ {1, . . . , n}, we have xkj ∈ Ann(bik). This
yields bikxkj = 0 for all i, j, k ∈ {1, . . . , n}, so BX = 0. Thus, we have proved that
N+(A) ⊆ N+(B). By swapping the roles of A in B we also get N+(B) ⊆ N+(A), so
N+(A) = N+(B). A similar argument yields that N−(A) = N−(B), thus A and B are
twin vertices.
Definition 3.5. Let S be a commutative semiring and let α ∈ S \ Z(S). We say that
α = e1 + e2 + · · · + es such that ei ̸= 0 for all i and eiej = 0 for all i ̸= j is a
decomposition of α of length s. The length ℓ(α) of α is the supremum of the length of a
decomposition of α (note that ℓ(α) can be infinite). We say that α is of maximal length if
ℓ(α) ≥ ℓ(β) for all β ∈ S \ Z(S).
A semiring S is decomposable if S \ Z(S) contains an element of length at least 2,
otherwise it is indecomposable.
Lemma 3.6. Let S be a commutative antiring and let α ∈ S \ Z(S) be of finite maximal
length s with decomposition α = e1 + e2 + · · · + es. Then, for every i ∈ {1, . . . , s}, the
subsemiring eiS is indecomposable.
Proof. Suppose that eiS is decomposable for some i ∈ {1, . . . , s}, say i = 1 without loss
of generality. By definition, there exists e1w ∈ e1S \ Z(e1S) such that e1w = f1 + f2,
where f1, f2 ∈ e1S \ {0} and f1f2 = 0. For all j ̸= 1, we have eje1w = 0 and thus
ejf1 = ejf2 = 0 by antinegativity. Let β = e1w + e2 + · · ·+ es.
Suppose that βx = 0 for some x ∈ S. By antinegativity, we have (e1w)(e1x) = 0 and
e2x = · · · = esx = 0. Since e1w is not a zero-divisor in e1S this implies that e1x = 0 and
therefore also αx = 0. However, α is not a zero-divisor, so we can conclude that x = 0.
This shows that β = f1 + f2 + e2 + · · · + es is not a zero-divisor in S, which is a
contradiction with the maximal length of α.
D. Dolžan et al.: The automorphism group of the zero-divisor digraph of matrices . . . 321
We shall investigate commutative antirings with identity where 1 is an element of finite
maximal length. The next lemma shows that in this case, we can study the automorphisms
of the zero-divisor digraph of the matrix ring componentwise.
Lemma 3.7. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1 + e2 + · · ·+ es. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))).
Then there exists ω ∈ Sym(s) such that, for every r ∈ {1, . . . , s}, we have σ(erMn(S)) =
eω(r)Mn(S).
Proof. Let r ∈ {1, . . . , s}, let i, j ∈ {1, . . . , n} and let B = σ(erEij). So, σ(erEij) =∑s
k=1 ekB. By Lemma 3.2, we have erEij =
∑s
k=1 σ
−1(ekB). Since S is antinegative,
for every k ∈ {1, . . . , s}, there exists fk ∈ S such that σ−1(ekB) = fkEij . Since
fk = fk(e1 + e2 + · · · + es) = fker ∈ erS, we have
∑s
k=1 fk = erz for z =
∑s
k=1 fk.
Observe that erEij = zEij yields z ∈ S \ Z(S).
Let k, k′ ∈ {1, . . . , s}. We have AnnL(fkEij),AnnL(fk′Eij) ⊆ AnnL(fkfk′Eij)
hence AnnL(ekB),AnnL(ek′B) ⊆ AnnL(σ(fkfk′Eij)). If k ̸= k′, then Mn(S) ⊆
AnnL(ekB) + AnnL(ek′B), which is possible only if fkfk′ = 0. We have shown that
fkfk′ = 0 for every k, k′ ∈ {1, . . . , s} with k ̸= k′.
It follows that
z = (e1 + e2 + · · ·+ es)z =
∑
i ̸=r
eiz +
s∑
k=1
fk
is a decomposition of z. Since z /∈ Z(S), eiz ̸= 0 and, since ℓ(z) ≤ ℓ(1) = s, it follows
that all but exactly one of the fk’s are 0. This implies that all but one of the ekB’s are 0
and there exists k ∈ {1, . . . , s} such that σ(erEij) = ekB. This shows the existence of a
permutation ω ∈ Sym(s) such that σ(erEij) = eω(r)B.
Let t ∈ {1, . . . , n}. We have e2r = er ̸= 0, so erEijerEjt ̸= 0 and (eω(r)B)σ(erEjt) ̸=
0. This implies σ(erEjt) ∈ eω(r)Mn(S).
As this holds for all j, t ∈ {1, . . . , n} and for any A ∈ Mn(S), we have A = (e1+e2+
· · ·+es)A, we have σ(erMn(S)) ⊆ eω(r)Mn(S). By the same token, we can conclude that
a twin vertex to a vertex from erMn(S) is itself in erMn(S), therefore also σ(erMn(S)) ⊆
eω(r)Mn(S). Since σ is a bijection, σ(erMn(S)) = eω(r)Mn(S).
We next focus on the automorphisms restricted to the matrices over indecomposable
subsemirings.
Proposition 3.8. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1 + e2 + · · · + es. Let u, v ∈ {1, . . . , s}, S1 = euS
and S2 = evS. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))) such that σ(Mn(S1)) = Mn(S2).
If i, j ∈ {1, . . . , n}, then there exist y ∈ S2 \ Z(S2) and k, ℓ ∈ {1, . . . , n} such that
σ(euEij) = yEkℓ.
Proof. Write σ(euEij) =
∑
k,ℓ βkℓEkℓ. Let Akℓ = σ
−1(βkℓEkℓ). By Lemma 3.2,
σ(euEij) and σ
(∑
k,ℓ Akℓ
)
are twin vertices, therefore euEij and
∑
k,ℓ Akℓ are twin ver-
tices as well. Now, twin vertices of euEij must be of the form zEij , so
∑
k,ℓ Akℓ = zEij
for some z ∈ S. Since z = z(e1 + e2 + · · · + es), we conclude that z ∈ S1. Note also
322 Ars Math. Contemp. 24 (2024) #P2.07 / 317–325
that z is not a zero-divisor in S1, since euEij and zEij are twin vertices. Since S is an-
tinegative, we can conclude that, for all k, ℓ ∈ {1, . . . , n}, there exist αkℓ ∈ S1 such that
Akℓ = αkℓEij and
∑
k,ℓ αkℓ = z.
Let k, k′, ℓ, ℓ′ ∈ {1, . . . , n} with (k, ℓ) ̸= (k′, ℓ′). Now, we either have k ̸= k′ or
ℓ ̸= ℓ′. Suppose first that k ̸= k′. Since S is commutative, we have AnnL(Akℓ) =
AnnL(αkℓEij) ⊆ AnnL(αkℓαk′ℓ′Eij). By Lemma 3.1, this implies AnnL(βkℓEkℓ) ⊆
AnnL(σ(αkℓαk′ℓ′Eij)). Similarly, we have AnnL(βk′ℓ′Ek′ℓ′) ⊆ AnnL(σ(αkℓαk′ℓ′Eij)).
Since k ̸= k′, Mn(S) = AnnL(βkℓEkℓ) + AnnL(βk′ℓ′Ek′ℓ′) ⊆ AnnL(σ(αkℓαk′ℓ′Eij)),
which implies σ(αkℓαk′ℓ′Eij) = 0 and thus αkℓαk′ℓ′ = 0. If ℓ ̸= ℓ′, we arrive at the same
conclusion by using right annihilators, namely that distinct αkℓ’s annihilate each other.
Since S1 is indecomposable by Lemma 3.6, the sum
∑
k,ℓ αkℓ = z has at most one non-
zero summand. It follows that there is at most one non-zero Akℓ and at most one non-zero
βkℓEkℓ. This concludes the proof of the first part, with y = βkℓ.
It remains to show that y /∈ Z(S2). Suppose, on the contrary, that y ∈ Z(S2). By the
first part of the result, there exist y′ ∈ S1 and i′, j′ ∈ {1, . . . , n} such that σ−1(evEkℓ) =
y′Ei′j′ . Since y ∈ Z(S2), we have AnnL(evEkℓ) ⊊ AnnL(yEkℓ) and AnnR(evEkℓ) ⊊
AnnR(yEkℓ). By Lemma 3.1, it follows that AnnL(y′Ei′j′) ⊊ AnnL(euEij) and of
course also AnnR(y′Ei′j′) ⊊ AnnR(euEij). This is only possible if i = i′ and j = j′
which implies AnnL(y′Eij) ⊊ AnnL(euEij), a contradiction.
Lemma 3.9. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1 + e2 + · · · + es. Let u, v ∈ {1, . . . , s}, S1 = euS
and S2 = evS. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))) such that σ(Mn(S1)) = Mn(S2). If
x ∈ Z(S1) and i, j ∈ {1, . . . , n}, then there exist z ∈ Z(S2) and k, ℓ ∈ {1, . . . , n} such
that σ(xEij) = zEkℓ.
Proof. By Proposition 3.8, we know that σ(euEij) = yEkℓ for some y /∈ Z(S2) and k, ℓ ∈
{1, . . . , n}. Since x ∈ Z(S1), we have AnnL(euEij) ⊊ AnnL(xEij) and AnnR(euEij) ⊊
AnnR(xEij). By Lemma 3.1, it follows that AnnL(yEkℓ) ⊊ AnnL(σ(xEij)) and also
AnnR(yEkℓ) ⊊ AnnR(σ(xEij)). This implies that all entries of σ(xEij) are zeros
except entry (k, ℓ), so σ(xEij) = zEkℓ for some z ∈ S2. Because AnnL(yEkℓ) ̸=
AnnL(σ(xEij)) = AnnL(zEkℓ) and AnnR(yEkℓ) ̸= AnnR(σ(xEij)) = AnnR(zEkℓ),
we must have z ∈ Z(S2).
Lemma 3.10. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1 + e2 + · · ·+ es. Let u, v ∈ {1, . . . , s}, S1 = euS and
S2 = evS. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))) such that σ(Mn(S1)) = Mn(S2). Then
there exists π ∈ Sym(n) such that σ(euEij) = evEπ(i)π(j) for all i, j ∈ {1, . . . , n}.
Proof. Let i, j, j′ ∈ {1, . . . , n} with j ̸= j′. By Proposition 3.8, there exist k, k′, ℓ, ℓ′ ∈
{1, . . . , n} such that σ(euEij) = evEkℓ and σ(euEij′) = evEk′ℓ′ . For all r, s ∈ {1, . . . , n}
with s ̸= i, we have euErs(euEij + euEij′) = 0. By Lemma 3.2, this implies that
σ(euErs)(euEkℓ + euEk′ℓ′) = 0 and thus
(∑
r,s ̸=i σ(euErs)
)
(euEkℓ + euEk′ℓ′) = 0.
By Proposition 3.8, σ(euErs) = evEr′s′ for some r′, s′ ∈ {1, . . . , n}. Since σ is a per-
mutation,
∑
r,s ̸=i σ(euErs) is a matrix with exactly n entries equal to 0. It follows that
k = k′.
By the paragraph above, there exists π ∈ Sym(n) such that σ(euEab) = evEπ(a)c, for
some c. A similar argument yields that there exists a permutation such that σ(euEab) =
D. Dolžan et al.: The automorphism group of the zero-divisor digraph of matrices . . . 323
evEcπ′(b), for some c. However, for every j, k ∈ {1, . . . , n} with j ̸= k, we have
EjjEkk = 0 and thus Eπ(j)π′(j)Eπ(k)π′(k) = 0. This implies that π(k) ̸= π′(j) for
every k ̸= j, so π(j) = π′(j). Therefore π′ = π.
For π ∈ Sym(n) and A ∈ Mn(S), let θπ(A) be the matrix obtained from A by applying
the permutation π to its rows and columns. Note that θπ induces a permutation of Mn(S).
Corollary 3.11. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1+e2+ · · ·+es. Let u, v ∈ {1, 2, . . . , s}, S1 = euS and
S2 = evS. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))) such that σ(Mn(S1)) = Mn(S2). Then
there exist π ∈ Sym(n) and τ an isomorphism from Γ(S1) to Γ(S2) such that, if we extend
τ entry-wise to a mapping Mn(S1) → Mn(S2) and restrict σ to Mn(S1), then σ = θπ ◦ τ .
Proof. By Lemma 3.10, there exists π ∈ Sym(n) such that σ(euEij) = θπ(evEij) for
all i, j ∈ {1, . . . , n}. Let ρ = θ−1π ◦ σ and note that ρ ∈ Aut(Γ(Mn(S))) and we have
ρ(euEij) = evEij for all i, j ∈ {1, . . . , n}.
Let x ∈ Z(S1) and i, j, j′ ∈ {1, . . . , n}. Clearly, ρ(Mn(S1)) = Mn(S2) so, by
Lemma 3.9, there exist z, z′ ∈ Z(S2) such that ρ(xEij) = zEij and ρ(xEij′) = z′Eij′ .
Let X = {s ∈ S2; sz = 0}. We show that X ⊆ Ann(z′). Let a ∈ S2 such
that az = 0. Note that (aEii)(zEij) = 0. Since a ∈ Z(S2), Lemma 3.9 implies that
there exists b ∈ Z(S1) such that aEii = ρ(bEii), hence ρ(bEii)ρ(xEij) = 0 and there-
fore also (bEii)(xEij) = 0 which implies bx = 0. It follows that (bEii)(xEij′) = 0,
ρ(bEii)ρ(xEij′) = 0 and (aEii)(z′Eij′) = 0 which yields az′ = 0.
We have shown that X ⊆ Ann(z′). A symmetrical argument yields Ann(z′) ⊆ X
hence X = Ann(z′) which implies that ρ(xEij′) = z′Eij′ = zEij′ (where for A ∈
Mn(S2), by a slight abuse of notation, A now refers to the image in Γ(Mn(S2)) and not in
Γ(Mn(S))). A similar argument shows that ρ(xEi′j) = zEi′j for all i′ ∈ {1, . . . , n}. This
implies that ρ(xEkℓ) = zEkℓ for all k, ℓ ∈ {1, . . . , n}.
Let τ denote the mapping S1 → S2 that satisfies ρ(xE11) = τ(x)E11. Since ρ is a
bijection from Mn(S1) to Mn(S2), τ is a bijection from S1 to S2. If x, y ∈ S1, then xy = 0
if and only if (xE11)(yE11) = 0 if and only if τ(x)τ(y) = 0, therefore τ is an isomorphism
from Γ(S1) to Γ(S2). Now, extend τ to an entry-wise mapping Mn(S1) → Mn(S2). Let
A = [aij ], B = [bij ] ∈ Mn(S1). Note that AB = 0 if and only if
∑n
k=1 aikbkj = 0 for
every 1 ≤ i, j ≤ n if and only if aikbkj = 0 for every 1 ≤ i, j, k ≤ n, so τ induces an
isomorphism from Γ(Mn(S1)) to Γ(Mn(S2)). Observe that, restricted to V(Γ(Mn(S1))),
we have ρ = τ . As σ = θπ ◦ ρ, this concludes the proof.
We can now join these findings into the following theorem.
Theorem 3.12. Let S be a commutative antiring and suppose 1 ∈ S is of finite maximal
length s with decomposition 1 = e1 + e2 + · · ·+ es. Let n ∈ N and σ ∈ Aut(Γ(Mn(S))).
Then there exist ω ∈ Sym(s) and, for every i ∈ {1, . . . , s}, there exist πi ∈ Sym(n)
and an isomorphism τi : Γ(eiS) → Γ(eω(i)S) such that, if we extend τi entry-wise to a
mapping Mn(eiS) → Mn(eω(i)S), then
σ(A) =
(
s∑
i=1
(θπi ◦ τi)(eiA)
)
for all A ∈ Mn(S).
324 Ars Math. Contemp. 24 (2024) #P2.07 / 317–325
Conversely, if ω ∈ Sym(s) has the property that, for every i ∈ {1, . . . , s}, we have
Γ(eiS) ∼= Γ(eω(i)S), τi is an isomorphism from Γ(eiS) to Γ(eω(i)S) and πi ∈ Sym(n),
then σ defined with σ(A) =
∑s
i=1 (θπi ◦ τi)(eiA) is an automorphism of Γ(Mn(S)).
Proof. By Lemma 3.7, there exists ω ∈ Sym(s) such that, for every i ∈ {1, . . . , s}, we
have σ(eiMn(S)) = eω(i)Mn(S).
By Corollary 3.11, there exist πi ∈ Sym(n) and τi an isomorphism from Γ(eiS) to
Γ(eω(i)S) such that, if we extend τi entry-wise to a mapping Mn(eiS) → Mn(eω(i)S) and
restrict σ to Mn(eiS), then σ = θπi ◦ τi.
Now, let A ∈ Mn(S). We have A = e1A+ e2A+ · · ·+ esA and the result follows by
Lemma 3.2.
Remark 3.13. Throughout the paper, we restricted ourselves to studying semirings with
the property that no non-zero-divisor element can be written as a sum of infinitely many
mutually orthogonal zero-divisors. Obviously, any semiring with a finite set of zero-
divisors satisfies this condition.
ORCID iDs
David Dolžan https://orcid.org/0000-0001-6548-3945
Gabriel Verret https://orcid.org/0000-0003-1766-4834
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.08 / 327–346
https://doi.org/10.26493/1855-3974.2947.cd6
(Also available at http://amc-journal.eu)
Quotients of skew morphisms of cyclic groups
Martin Bachratý *
Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia
Received 25 August 2022, accepted 19 April 2023, published online 4 October 2023
Abstract
A skew morphism of a finite groupB is a permutation φ ofB that preserves the identity
element of B and has the property that for every a ∈ B there exists a positive integer ia
such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms
for all finite cyclic groups is notoriously hard, with no such classification available up to
date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|,
called the quotient of φ. In this paper, we use this relationship and other observations
to prove new theorems about skew morphisms of finite cyclic groups. In particular, we
classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m
odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic
groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms
for cyclic groups of order up to 161.
During the preparation of this paper we noticed a few flaws in Section 5 of the paper
Cyclic complements and skew morphisms of groups from 2016. We propose and prove
weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6
and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs
(in the aforementioned paper) that use at least one of those problematic assertions.
Keywords: Skew morphism, cyclic group, coset-preserving, quotient, square-free.
Math. Subj. Class. (2020): 20B25, 05C25, 05E18
1 Introduction
A skew morphism of a finite group B is a permutation φ of B that preserves the identity
element of B and has the property that for every a ∈ B there exists a positive integer ia
such that φ(ab) = φ(a)φia(b) for all b ∈ B. The order of a skew morphism, denoted by
*The author acknowledges the use of the MAGMA system [7] to find examples of skew morphisms relevant to
this paper. The author also acknowledges support from the APVV Research Grants 17-0428 and 19-0308, and the
VEGA Research Grants 1/0206/20 and 1/0567/22.
E-mail address: martin.bachraty@stuba.sk (Martin Bachratý)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
328 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
ord(φ), is defined as the order of the cyclic group ⟨φ⟩. Note that for each a ∈ B there
is a unique choice for ia such that ia ∈ {1, 2, . . . , ord(φ) − 1} (unless φ is the identity
permutation). The function π that maps each element a ∈ B to this integer ia is called the
power function of φ, and it satisfies φ(ab) = φ(a)φπ(a)(b) for all a, b ∈ B. In the case
when φ is the identity permutation of B, we define π(a) = 1 for all a ∈ B.
Skew morphisms were first introduced by Jajcay and Širáň in [18], with primary inter-
est in their connection to the regular Cayley maps. Skew morphisms are also intriguing
from a purely group-theoretical point of view, mainly due to their close relationship with
group automorphisms, with which they share a number of important features. The prob-
lem of classifying all skew morphisms for given families of finite groups has gained much
attention in the last two decades; see [8, 10, 23, 24] for example. Recently, in [4], skew
morphisms were classified for all finite simple groups, and we understand that a classifica-
tion for dihedral groups is imminent; see [19]. On the other hand, the problem of finding
all skew morphisms for finite cyclic groups remains open, despite recent positive progress,
which we discuss next.
Automorphisms of a finite group B are special cases of skew morphisms (with π(a) =
1 for all a ∈ B), and as such can be viewed as an important family of skew morphisms.
There are also other intriguing families of skew morphisms, for example, coset-preserving
(sometimes also called smooth) skew morphisms, which are defined as skew morphisms
satisfying π(a) = π(φ(a)) for all a ∈ B. Coset-preserving skew morphisms have been
fully classified for all finite cyclic groups in [6]. Another interesting family of skew mor-
phisms that is fully understood for finite cyclic groups consists of all skew morphisms φ
such that φ2 is an automorphism of the same group; see [16].
While there is no classification of skew morphisms of finite cyclic groups available
to date, skew morphisms have been fully classified for some specific (infinite) families of
finite cyclic groups. Most notably, this was done for cases where the order of a cyclic
group is a prime [18] (in this case, all skew morphisms are automorphisms), a product of
two distinct primes [20], and any power of an odd prime [21]. Some partial progress for
cyclic 2-groups can be found in [14].
Another approach for studying skew morphisms of finite cyclic groups is to find a
connection between skew morphisms of a given cyclic group B and skew morphisms of
cyclic groups of smaller orders. Presumably the strongest finding to date made in this
direction is the observation of Kovács and Nedela proved in [20] which states that if
gcd(m,n) = gcd(m,ϕ(n)) = gcd(ϕ(m), n) = 1, then the skew morphisms of Zmn
are exactly the direct products of skew morphisms of Zm and Zn. There is also a useful
connection between general skew morphisms and coset-preserving skew morphisms for fi-
nite cyclic groups. Namely, for each skew morphism φ of a finite cyclic group B there
exists an exponent e such that φe is a non-trivial coset-preserving skew morphism of B;
see [5].
In this paper, we combine a number of known facts about skew morphisms (which we
summarise in Sections 2 and 3) with new observations presented in Section 4, to prove a
number of theorems about skew morphisms of cyclic groups. Namely, in Section 5 we
develop a new method for finding skew morphisms of cyclic groups, and implement it to
obtain a census of all skew morphisms of cyclic groups of order up to 161. (Up to the
time of writing this paper, and apart from some specific orders, skew morphisms of cyclic
groups were known only up to order 60; see [9].) Further, in Section 6 we show that all skew
morphisms of Zn are coset-preserving if and only if n = 2em for some e ∈ {0, 1, 2, 3, 4}
M. Bachratý: Quotients of skew morphisms of cyclic groups 329
and m odd and square-free. As a consequence, we obtain a complete classification of skew
morphisms for all cyclic groups of order expressible in this form, significantly expanding
the list of finite cyclic groups for which such classification is available. During the review
process, it was communicated to us that Kan Hu, István Kovács and Young Soo Kwon has
recently submitted a paper devoted to similar ideas to those we investigated in Section 6 of
this paper.
2 Preliminaries
In this section, we recall some definitions from group theory and provide some background
from the theory of skew morphisms. All groups considered in this paper are assumed to be
finite. For the cyclic group of order n we use the additive notation Zn, and so the elements
of Zn may be viewed as integers in the interval [0, n− 1]. We also let Sym(G) denote the
symmetric group on (the underlying set of) a group G.
The core of a subgroup H in a group G is the largest normal subgroup of H contained
in G. We say that H is core-free in G if the core of H in G is trivial. A complement for H
in G is a subgroup K of G such that G = HK and H ∩K = {1}. The following theorem
proved by Lucchini in [22] will be helpful.
Theorem 2.1 ([22]). Let C be a cyclic proper subgroup of a group G. If C is core-free in
G, then |C| < |G : C|.
Next, let φ be a skew morphism of a group B, and identify B with the subgroup of
Sym(B) which acts by left multiplication. Then it can be easily checked that B⟨φ⟩ is a
subgroup of Sym(B) (see [20] for example). Moreover, B⟨φ⟩ is a complementary factori-
sation and ⟨φ⟩ is core-free in B⟨φ⟩ (see [12, Lemma 4.1]). A group G containing B which
has a cyclic core-free complement C for B is called a skew product group for a group B,
and we say that C is a skew complement (for B in G). The skew product group B⟨φ⟩ (for
B) with skew complement ⟨φ⟩ described in this paragraph is said to be induced by φ.
Conversely, let G be a skew product group for a group B, and let c be a generator of a
skew complement for B in G. Note that every element g ∈ G is uniquely expressible in a
form g = ac′ with a ∈ B and c′ ∈ C. Then for every a ∈ B there exists a unique a′ ∈ B
and a unique exponent j ∈ {1, 2, . . . , |C| − 1} such that ca = a′cj , and this induces a
bijection φ : B → B and a function π : B → N, defined by φ(a) = a′ and π(a) = j. It
can be easily checked that φ is a skew morphism of B with power function π. We say that
φ is induced by the pair (B, c).
Recall that if φ is a skew morphism of B with power function π, then we have φ(ab) =
φ(a)φπ(a)(b) for all a, b ∈ B. (Hence, if B = Zn, then φ(a+ b) = φ(a)+φπ(a)(b) for all
a, b ∈ Zn.) Also recall that an automorphism of B is a skew morphism with π(a) = 1 for
all a ∈ B. In what follows, it will be often convenient to distinguish between general skew
morphisms and skew morphisms that are not automorphisms, and so we will refer to the
latter as proper skew morphisms. We say that φ is trivial if it is the identity permutation
of B. The kernel of φ, denoted by kerφ, is the subset {a ∈ B | π(a) = 1} of B. By
definition, φ is an automorphism of B if and only if kerφ = B. In the case of proper skew
morphisms kerφ is not equal to B, but it is always a subgroup of B; see [18, Lemma 4].
Since kerφ is a subgroup of B and also φ(ab) = φ(a)φ(b) for all a, b ∈ kerφ, it
follows that φ restricts to a group isomorphism from the kernel to its image. In particular,
if kerφ is preserved by φ set-wise, then φ restricts to an automorphism of kerφ. In [11]
this was shown to always be true for abelian groups. We also have the following.
330 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
Lemma 2.2 ([18]). Let φ be a skew morphism of a group B with power function π. Then
two elements a, b ∈ B belong to the same right coset of the subgroup kerφ in B if and
only if π(a) = π(b).
Theorem 2.3 ([12]). Every skew morphism of a non-trivial group has non-trivial kernel.
An immediate consequence of Theorem 2.3 is that every skew morphism of a group
of prime order is an automorphism. The following facts about kernels of skew morphisms
will be useful, too.
Lemma 2.4 ([12]). Let G be a skew product group for a group B with skew complement
C, and let c be a generator of C. If φ is the skew morphism induced by (B, c), then kerφ is
the largest subgroup H of B for which cHc−1 ⊆ B. In particular, φ is an automorphism
of B if and only if B is normal in G.
Proposition 2.5 ([12]). Let φ be a skew morphism of a group B, and let N be a subgroup
of kerφ that is normal in B and preserved by φ. Then the mapping φ∗N : B/N → B/N
given by φ∗N (x) = Nφ(x) is a well-defined skew morphism of B/N .
Lemma 2.6 ([12]). Let φ be a skew morphism of a finite abelian group A with power
function π. Also suppose that N is any non-trivial subgroup of kerφ preserved by φ, let i
be the exponent of N , and let φ∗N be the skew morphism of A/N induced by φ. If a is an
element of A such that Na lies in the kernel of φ∗N , then iπ(a) ≡ i (mod ord(φ)) and, in
particular, if gcd(i, ord(φ)) = 1, then a ∈ kerφ.
Note that if φ is a skew morphism of Zn, then kerφ is normal in Zn and φ restricts to
an automorphism of kerφ. Moreover, since Zn is cyclic, so is (its subgroup) kerφ. Noting
that every automorphism of a cyclic group preserves all of its subgroups, we deduce that
φ preserves all subgroups of kerφ. Hence it follows from Proposition 2.5 that φ∗N is a
well defined skew morphism of Zn/N for each subgroup N of kerφ. In the case when
N = kerφ, we will write simply φ∗ instead of φ∗N .
We proceed with some useful facts about orders of skew morphisms.
Proposition 2.7 ([20]). Let φ be a skew morphism of a group B, and let T be an orbit of
⟨φ⟩. If ⟨T ⟩ = B, then ord(φ) = |T |.
Theorem 2.8 ([12]). The order of a skew morphism of a non-trivial group B is less than
the order of B.
Theorem 2.9 ([20]). If φ is a skew morphism of a group Zn, then ord(φ) is a divisor of
nϕ(n). Moreover, if gcd(ord(φ), n) = 1, then φ is an automorphism of Zn.
The periodicity of a skew morphism φ of a group B, denoted by pφ, is the smallest
positive integer such that π(a) = π(φpφ(a)) for all a ∈ B. Similarly, the periodicity of a ∈
B (with respect to φ) is the smallest positive integer pa such that π(a) = π(φpa(a)). Note
that if kerφ is preserved by φ (which is always true if B is abelian), then the periodicity of
φ can be defined equivalently as the order of φ∗.
Recall that φ is coset-preserving if π(a) = π(φ(a)) for all a ∈ B. Equivalently, coset-
preserving skew morphisms can be viewed as skew morphism that preserves all right cosets
of kerφ inB, or as skew morphisms with the periodicity equal to 1. The following theorem
about periodicities underlines the importance of coset-preserving skew morphisms in the
study of skew morphisms of abelian (and, in particular, cyclic) groups.
M. Bachratý: Quotients of skew morphisms of cyclic groups 331
Theorem 2.10 ([5]). If φ is a skew morphism of an abelian group A, then φpφ is a coset-
preserving skew morphism of A. Moreover, if A is cyclic, b is a generator of A, and φ is
non-trivial, then pφ = pb and pφ < ord(φ).
The following fact will be helpful too.
Lemma 2.11 ([12]). Let φ be any skew morphism of a finite group G, and let H be any
finite group. Then φ can be extended to a skew morphism θ of G×H , such that θ ↾G= φ
and ker θ = kerφ×H .
We conclude this section with two well-known facts about skew morphisms that are
easy exercises, but we include their proofs for completeness.
Lemma 2.12. Let φ be a skew morphism of an abelian group A. If φ(a) = a for some
a ∈ A, then a ∈ kerφ.
Proof. Let a′ be any element of A. Since a is fixed by φ, we have φ(aa′) = aφπ(a)(a′).
On the other hand, we have φ(aa′) = φ(a′a) = φ(a′)φπ(a
′)(a) = φ(a′)a, and hence
φ(a′) = φπ(a)(a′) for all a′ ∈ A. It follows that π(a) = 1, and therefore a ∈ kerφ.
Lemma 2.13. Let φ be a skew morphism of a group B with power function π, and let
a ∈ B. Then:
φ(ai) = φ(a)φπ(a)(a)φπ(a
2)(a) . . . φπ(a
i−1)(a) for all i ∈ N.
Proof. The assertion is trivially true for i = 1. Next, if it holds for some positive integer
j, then φ(aj+1) = φ(aja) = φ(aj)φπ(a
j)(a) = φ(a)φπ(a)(a) . . . φπ(a
j−1)(a)φπ(a
j)(a),
and so it is also true for j + 1. Hence the proof follows by induction.
3 Skew morphisms of abelian groups
Several useful facts about skew morphisms of abelian groups (which we also apply in this
paper) were proved in [12]. During the preparation of this paper, however, we noticed
that three assertions in [12] do not hold. In this section, we list the incorrect findings and
provide a counterexample for each of them. We also propose and prove weaker versions
of the original statements. Finally, we discuss all proofs in [12] that use at least one of the
flawed statements, and show that in all cases it is sufficient to replace the flawed statements
by our modifications. For the rest of the section, we let φ be a skew morphism of an abelian
group A with power function π. Also, to distinguish between references to this paper and
references to [12], we put an asterisk after each numbered reference in the latter case.
The first flawed assertion in [12] is part (b) of Lemma 5.3∗. It states that if N is a non-
trivial subgroup of kerφ preserved by φ, and φ is not an automorphism of A, then ord(φ)
has a non-trivial divisor in common with the exponent of N . To show that this is not true,
note that according to [9] the cyclic group of order 12 admits a proper skew morphism
ψ of order 3 with kernel (which is preserved by ψ) of order 6. Since kerψ is cyclic and
preserved by ψ, so is its unique subgroup of order 2. But the exponent of this subgroup,
which is 2, does not have a non-trivial divisor in common with ord(ψ). We propose the
following modification:
Lemma 3.1. If φ is a proper skew morphism of an abelian group A, then ord(φ) has a
non-trivial divisor in common with the exponent of kerφ.
332 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
Proof. Let K = kerφ, let e be the exponent of K, and let L/K be the kernel of the skew
morphism φ∗ of A/K induced by φ. Since φ is proper we know that A/K is a non-trivial
group, and by Theorem 2.3 it follows that L/K is non-trivial. In particular, there exists an
element a of A such that a /∈ K and a ∈ L. It follows that π(a) ̸≡ 1 (mod ord(φ)), by
Lemma 2.6 we have eπ(a) ≡ e (mod ord(φ)), and the rest follows.
Part (b) of Lemma 5.3∗ is used in the proofs of six theorems presented in [12]. Proofs of
Theorem 5.4∗, Theorem 5.10∗, Theorem 6.2∗ and Theorem 6.4∗ are all easily fixable, since
in each case Lemma 5.3(b)∗ is applied for N = K, and hence it is sufficient to replace it
with Lemma 3.1. The proof of Theorem 6.1∗ can also be corrected. Here Lemma 5.3(b)∗ is
used to show that if φ is proper and A ∼= Zn, then gcd(ord(φ), n) ̸= 1. Since A is cyclic,
the exponent of kerφ is equal to the order of kerφ (which necessarily divides n), so this
is an easy consequence of Lemma 3.1. Finally, since Theorem 7.3∗ was already proved
previously in [20], there is no need to fix its alternative proof presented in [12]; although
we believe that it is possible.
Another flawed assertion in [12] is Theorem 5.6∗. It states that if L/(kerφ) is the
kernel of the skew morphism φ∗ of A/(kerφ) induced by φ, and p is a prime that divides
|L| but not |kerφ|, then p < q for every prime divisor q of |kerφ|. Again, we provide
a counterexample that shows that this is not true. According to [9] the cyclic group of
order 42 admits a proper skew morphism ρ of order 7 with kernel of order 14. Since
Z42/(ker ρ) is isomorphic to Z3 and Z3 does not admit any proper skew morphism, it
follows that the kernel L/(ker ρ) (of the skew morphism of Z42/(ker ρ) induced by ρ) is
equal to Z42/(ker ρ). Therefore, |L| = |Z42| and, in particular, 3 divides |L|. But 3 is
greater than 2, and 2 is a prime divisor of |ker ρ|. We propose the following modification:
Theorem 3.2. Let φ be a skew morphism of the finite abelian group A, let K = kerφ, and
let q be any prime divisor of |K|. Also let N be a subgroup of K consisting of the identity
and all elements of order q, and let L/N be the kernel of the skew morphism φ∗N of A/N
induced by φ. If p is a prime that divides |L| but not |K|, then p < q.
Proof. Suppose that such a prime p exists. Since K is abelian, we know that N is a
subgroup of K of exponent q that is invariant under φ. Next, let a be any element of
order p in L, let m = ord(φ), and let π be the power function of φ. Since L/N is the
kernel of φ∗N , we know by Lemma 2.6 that q(π(a) − 1) ≡ 0 (mod m). If q is relatively
prime to m, then π(a) ≡ 1 (mod m) and so a ∈ K, which is impossible since K has
no element of order p. Thus q divides m and π(a) − 1 ≡ 0 (mod m/q). In particular,
π(a) = 1 + i(m/q) where 1 ≤ i ≤ q − 1, so there are at most q − 1 possibilities for π(a).
The same holds for every non-trivial power of a. So now if p > q, then by the pigeon-
hole principle two different powers of a will have the same value under π, in which case
they lie in the same coset of N . But that cannot happen since K ∩ ⟨a⟩ is trivial. Thus
p < q.
The only application of the flawed original version of Theorem 5.6∗ is Corollary 5.7∗.
This final problematic assertion in [12] states that every prime divisor of |kerφ| is greater
than every prime that divides |A| but not |kerφ|. To see that this is not true, take the skew
morphism ρ of Z42 with kernel of order 14 discussed earlier. Since 2 divides |ker ρ| and 3
divides |Z42| but not |ker ρ|, the statement is clearly not true. The following, however, still
holds.
M. Bachratý: Quotients of skew morphisms of cyclic groups 333
Corollary 3.3. Let A be a non-trivial finite abelian group, and let p be the largest prime
divisor of |A|. Then the order of the kernel of every skew morphism of A is divisible by p
when p is odd, or by 4 when p = 2.
Proof. Let φ be any skew morphism of A, let K = kerφ, and suppose to the contrary that
p does not divide |K|. Also let q be any prime divisor of |K|, let N be a subgroup of K
consisting of the identity and all elements of order q, and let L/N be the kernel of the skew
morphism φ∗N of A/N induced by φ. If p divides |L/N |, then by Theorem 3.2 we know
that p is smaller than q, so this cannot happen. It follows that p does not divide |L/N |, so
we can repeat the same argument for the skew morphism φ∗N of A/N with kernel L/N .
(Note that p divides |A/N |.) Since A is finite, this will eventually terminate for some
groups A′, N ′ and L′ with |L′/N ′| = 1. Then since the kernel L′/N ′ is trivial, it follows
by Theorem 2.3 thatA′/N ′ is a trivial group, and hence |A′| = |N ′|. But this is impossible,
since p divides |A′| but not |N ′|. The second part for p = 2 follows from the original proof
of [12, Corollary 5.7].
Both applications of Corollary 5.7∗, namely the proofs of Theorem 6.2∗ and Theo-
rem 9.1∗, only use the fact that the order of kerφ is divisible by the largest prime divisor
of |A|. Since this follows by Corollary 3.3, both theorems still hold, and their proofs can
be corrected by minor changes in their wording.
4 Quotients and their properties
In this section, we will show that if BC is a complementary product of two cyclic groups
with C core-free in BC, then not only C corresponds to some skew morphism of B (of
order |C|), but also B corresponds to some skew morphism of C (of order smaller than
|B|). Let φ be a skew morphism of a cyclic group B, let G = B⟨φ⟩, and let C = ⟨φ⟩.
Also let b be a generator of B. To distinguish between φ as a permutation of B and φ as a
generator of the cyclic group C, we use c in the latter case. Since C is core-free in G, by
Theorem 2.1 we have
|G : B| = |C| < |G : C| = |B| ,
so (again by Theorem 2.1) we find that B has a non-trivial core in G. Let K denote the
core of B in G, and for every X ≤ G let X denote XK/K (∼= X/(X ∩ K)). Next, let
H be a subgroup of B such that cHc−1 ⊆ B. Then, since B is cyclic, H is the unique
subgroup of B of order |H|, and so cHc−1 = H . It follows that H is normal in G, and so
it is contained in K. Moreover, since K is the core of B in G, we have cKc−1 = K ⊆ B,
and hence by Lemma 2.4 we find that K = kerφ.
Now we look closely at the product G = B C. First, noting that K ∩ C = {1} we
have C ∼= C (and so C is cyclic, and hence abelian) and B ∩ C = {1}. Since B is
cyclic, so is its quotient B, and by the definition of K we deduce that B is core-free in G.
Now it is straightforward to check that the bijection that maps every element d ∈ C to the
unique element d′ ∈ C such that d b = bjd′ defines a skew morphism of C, with power
function π given by π(d) = j. (We choose b to be the image of b ∈ B under the natural
homomorphism from B to B; since b is a generator of B, it follows that b is a generator of
B.)
A skew morphism of C (∼= C) constructed in the way described in the previous para-
graph is called the quotient of φ (with respect to b), and will be denoted by φ. Since a
cyclic group B can be generated by different elements, φ can have more than one quotient.
334 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
(In fact, by [4, Remark 5.1] this is always true unless B has a unique generator.) In the
case of additive notation B = Zn, and unless otherwise specified, by the quotient of φ we
understand the quotient with respect to 1.
The above construction (proposed in the author’s PhD thesis [3]) was also introduced
independently in [15], where it was noted that if φ is a skew morphism of Zn and φ is a quo-
tient of φ, then (φ,φ) is an (ord(φ), n)-reciprocal pair of skew morphisms. A pair (φ, ρ) of
skew morphisms of Zn and Zm with power functions π and τ is called (m,n)-reciprocal if
ord(φ) dividesm, ord(ρ) divides n, and the congruences π(i) ≡ ρi(1) (mod ord(φ)) and
τ(j) ≡ φj(1) (mod ord(ρ)) hold for each i ∈ Zn and j ∈ Zm. We note that while every
pair (φ,φ) gives an (ord(φ), n)-reciprocal pair of skew morphisms, not every reciprocal
pair arises in this way. For example, there exist (m,n)-reciprocal pairs of skew morphisms
with m = n, but by Theorem 2.8 we know that ord(φ) is always strictly smaller than n.
The following observation is an easy consequence of the fact that a skew morphism (of
a cyclic group) and its quotient always give a reciprocal pair of skew morphisms.
Lemma 4.1. Let φ be a skew morphism of Zn with power function π, and let φ be the
quotient of φ with power function π. Then for every i ∈ N:
(a) π(i) = φ i(1), so in particular ord(φ) = n/|kerφ|; and
(b) φi(1) ≡ π(i) (mod n/|kerφ|).
Proof. First, since (φ,φ) is an (ord(φ), n)-reciprocal pair of skew morphisms, we have
π(i) ≡ φ i(1) (mod ord(φ)). Hence, since both π and φ are mappings into Zord(φ), it
follows that π(i) = φ i(1). The second part of (a) follows from the fact that n/|kerφ|
is the smallest non-zero integer in kerφ. Finally, (b) follows easily as π(i) ≡ φi(1)
(mod ord(φ)) and ord(φ) = n/|kerφ|.
Next we provide a lemma which shows that quotients can be used to check whether a
skew morphism of a cyclic group is an automorphism or a coset-preserving skew morphism.
Lemma 4.2. A skew morphism φ of Zn is coset-preserving if and only if the quotient φ of
φ is an automorphism. Moreover, φ is proper if and only if φ is non-trivial.
Proof. Let φ be a coset-preserving skew morphism of Zn. This is equivalent with φ(1) ≡ 1
(mod n/|kerφ|), which by Lemma 4.1(b) happens if and only if π(1) = 1. This proves
the first part. The second part follows easily by Lemma 4.1(a) as φ ∈ Aut(Zn) if and only
if π(1) = 1, and φ is trivial if and only if φ(1) = 1.
We note that a part of the Lemma 4.2 was proved in [17], where it was shown that if φ
is an automorphism, then φ is coset-preserving, but not the other way around. Also note
that since the only skew morphism of Z2 is the identity mapping, it follows immediately
from Lemma 4.2 that if a skew morphism of a cyclic group has order 2, then it must be an
automorphism. (This is also an easy consequence of Lemma 2.4.) Another consequence of
Lemma 4.2 is the fact that a proper skew morphism of Zn is coset-preserving if and only
if the quotient of its quotient is the identity mapping. Somewhat interestingly, Lemma 4.2
also implies that by taking quotients every skew morphism of a cyclic group can be reduced
to a non-trivial automorphism of the same group.
M. Bachratý: Quotients of skew morphisms of cyclic groups 335
5 Using quotients to generate skew morphisms
In this section, we describe an algorithm for finding recursively skew morphisms of cyclic
groups based on various observations about the quotients of skew morphisms.
5.1 Skew morphisms with a given quotient
First, we explain how to find all skew morphisms of a cyclic group with a given quotient.
The following observation about quotients of skew morphisms of cyclic groups will be
useful.
Proposition 5.1. Let φ be a skew morphism of Zn with power function π, and let φ be the
quotient of φ with power function π. Then:
(a) the periodicity pφ of φ is the smallest generator of kerφ;
(b) ord(φ) = |kerφ|pφ; and
(c) the orbit T of ⟨φ⟩ that contains 1 is expressible in the form
T = (x1, . . . , xpφ , ψ(x1), . . . , ψ(xpφ), . . . , ψ
ord(ψ)−1(x1), . . . , ψ
ord(ψ)−1(xpφ)),
(5.1)
for some coset-preserving skew morphism ψ of Zn such that ord(ψ) = ord(φ)/pφ
andψ(1) ≡ 1 (mod n/|kerφ|), and with x1 = 1 and xi ≡ π(i−1) (mod n/|kerφ|)
for each i ∈ {2, . . . , pφ}.
Proof. First, by Theorem 2.10 we have pφ = p1. Then, since the values taken by π
at any two elements of Zn are equal if and only if they belong to the same right coset
of kerφ in Zn, it follows that p1 is the smallest positive integer such that 1 ≡ φp1(1)
(mod n/|kerφ|), which by Lemma 4.1(b) is equivalent with 1 ≡ π(p1) (mod n/|kerφ|).
Noting that n/|kerφ| = ord(φ), we deduce that p1 is the smallest positive integer such
that π(p1) = 1, and (a) follows. Moreover, since pφ is the smallest non-trivial element of
kerφ, which is a subgroup of Zord(φ), it follows that ord(φ) = |kerφ|pφ, which proves
(b).
To prove the final assertion, let T denote the orbit of ⟨φ⟩ that contains 1, and let
ψ = φp1 . By Theorem 2.10 we know that ψ is a coset-preserving skew morphism of
Zn, and by the definition of the periodicity we have ψ(1) ≡ 1 (mod n/|kerφ|). By
Proposition 2.7 we know that the size of T is equal to ord(φ). Moreover, p1 divides |T |
(see [6, Lemma 3.1]), and hence ord(ψ) = ord(φp1) = ord(φ)/p1, and the effect of ψ
on T induces p1 cycles, each of length ord(φ)/p1. Finally, by Lemma 4.1(b) we have
xi = φ
i−1(1) ≡ π(i− 1) (mod n/|kerφ|) and the rest follows.
Let ρ be a skew morphism of a cyclic group Zm. We will provide a detail explanation
of the method for finding all skew morphisms φ of Zn with quotient ρ.1
First, we find the smallest positive integer j such that j ∈ ker ρ. Then by Propo-
sition 5.1(a) we have p1 = pφ = j. Next, since ρ is a quotient of φ, it follows by
1It is important to emphasise here that this method finds all skew morphisms with a particular quotient only for
a given cyclic group. If we do not restrict ourselves to a specific group, then in some cases we can find infinitely
many skew morphisms with a given quotient. For example, using the classification of skew morphisms for cyclic
p-groups of odd order (presented in [21]) it can be shown that each cyclic 3-group of order at least 9 admits a
skew morphism whose quotient is the skew morphism (1, 3, 5) of Z6.
336 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
Lemma 4.1(a) that ord(ρ) = n/|kerφ|, and hence kerφ is the unique subgroup of Zn of or-
der n/ord(ρ). As a next step we find all coset-preserving skew morphisms ψ of Zn satisfy-
ing ord(ψ) = m/pφ andψ(1) ≡ 1 (mod n/|kerφ|) (note here thatm = |Zm| = ord(φ)).
This allows us to identify all possible candidates for the orbit T of ⟨φ⟩ that contains 1,
using (5.1). (For each choice of ψ, we have at most |kerφ|p1−1 = (n/ord(ρ))p1−1 candi-
dates; the number of candidates could be smaller since sometimes (5.1) does not define a
cycle on B.)
Next, suppose that φ is a skew morphism, and let φ1 be the cyclic permutation of T
induced by φ. Then by Lemma 4.1(a) we have π(i) = ρi(1), and hence by Lemma 2.13
we find that
φ(i) = φ1(1) + φ
ρ(1)
1 (1) + φ
ρ2(1)
1 (1) + · · ·+ φ
ρi−1(1)
1 (1) for all i ∈ N. (5.2)
As a final step, for each candidate for φ1 we use (5.2) to define a function φ : Zn → Zn,
and then check whether φ is a skew morphism of Zn, and T an orbit of ⟨φ⟩. It can be easily
verified that if this is true, then ρ is the quotient of φ.
Remark 5.2. Note that to use (5.2), we do not need to know the complete orbit T , but only
the elements φ ρ
i(1)
1 (1) for each i ∈ {0, 1, . . . , ord(ρ)}. In fact, since for every positive
integer j we have φ j+p11 (1) = ψ(φ
j
1 (1)), only elements of the form φ
e
1 (1) with e ≡ ρi(1)
(mod p1) are needed to define φ. In some cases, this significantly reduces the number of
possible candidates for φ1.
5.2 Algorithm for finding all skew morphisms of a cyclic group
We are ready to describe our algorithm for finding all skew morphisms of cyclic groups
up to any order. This algorithm is recursive in the sense that it takes the sets of all skew
morphisms of the groups Zm for m ∈ {2, 3, . . . , n − 1} as input and outputs all skew
morphisms of Zn. Since the only skew morphism of Z2 is the identity permutation, the
algorithm can be easily initialised.
As the first step, we use the method presented in [6] to find all coset-preserving skew
morphisms of Zn. Further details on this method are available in Section 6.3. Next, let
φ be a skew morphism of Zn that is not coset-preserving, and let φ be the quotient of φ.
Then by Lemma 4.2 we know that φ is a proper skew morphism of Zord(φ). Moreover, by
Theorem 2.8 and Theorem 2.9 we find that ord(φ) < n, and that ord(φ) divides nϕ(n),
and gcd(ord(φ), n) ̸= 1. Since we know all skew morphisms of Zm for each m < n, and
we also know all coset-preserving skew morphisms of Zn, it follows that we can simply
apply the method explained in the previous subsection to find all skew morphisms of Zn
that are not coset-preserving. Together with the coset-preserving skew morphisms of Zn
(which include all automorphisms and were found earlier), this gives all skew morphisms
of Zn.
A MAGMA [7] implementation of the described algorithm succeeded in finding all skew
morphisms of cyclic groups of order up to 161. (The file listing all of these skew morphisms
is available at [2].) This significantly improves the previous largest complete list [9] which
goes up to the order 60. In Table 1 we summarise the information obtained about skew
morphisms of cyclic groups Zn for n ≤ 161. We include a group in the table if and
only if it admits a proper skew morphism. Moreover, if a listed group admits a proper
skew morphism that is not coset-preserving, then the order of the group is preceded by
M. Bachratý: Quotients of skew morphisms of cyclic groups 337
n Skew Classes n Skew Classes n Skew Classes
6 2 + 2 1 58 28 + 28 1 114 148 + 36 7
8 2 + 4 1 60 80 + 16 17 116 112 + 56 3
∗9 4 + 6 2 62 30 + 30 1 ∗117 88 + 72 11
10 4 + 4 1 ∗63 44 + 36 7 118 58 + 58 1
12 4 + 4 2 ∗64 268 + 32 42 120 208 + 32 43
14 6 + 6 1 66 60 + 20 13 ∗121 900 + 110 90
16 12 + 8 4 68 64 + 32 3 122 60 + 60 1
∗18 24 + 6 6 70 72 + 24 11 124 60 + 60 2
20 16 + 8 3 ∗72 156 + 24 36 ∗125 1568 + 100 152
21 12 + 12 1 74 36 + 36 1 ∗126 348 + 36 34
22 10 + 10 1 ∗75 96 + 40 24 ∗128 1132 + 64 114
24 16 + 8 7 76 36 + 36 2 129 84 + 84 1
∗25 48 + 20 12 78 104 + 24 9 130 144 + 48 17
26 12 + 12 1 80 152 + 32 26 132 120 + 40 26
∗27 64 + 18 20 ∗81 676 + 54 110 134 66 + 66 1
28 12 + 12 2 82 40 + 40 1 ∗135 256 + 72 80
30 24 + 8 7 84 104 + 24 14 136 228 + 64 10
∗32 60 + 16 14 86 42 + 42 1 138 132 + 44 25
34 16 + 16 1 88 80 + 40 15 140 240 + 48 29
∗36 48 + 12 12 ∗90 216 + 24 36 142 70 + 70 1
38 18 + 18 1 92 44 + 44 2 ∗144 552 + 48 96
39 24 + 24 1 93 60 + 60 1 146 72 + 72 1
40 44 + 16 9 94 46 + 46 1 ∗147 960 + 84 68
42 52 + 12 7 ∗96 272 + 32 58 148 144 + 72 3
44 20 + 20 2 ∗98 480 + 42 38 ∗150 648 + 40 74
∗45 16 + 24 8 ∗99 40 + 60 20 152 144 + 72 23
46 22 + 22 1 ∗100 512 + 40 42 ∗153 64 + 96 32
48 64 + 16 20 102 96 + 32 19 154 180 + 60 17
∗49 180 + 42 30 104 132 + 48 13 155 120 + 120 1
∗50 152 + 20 18 105 48 + 48 4 156 352 + 48 22
52 48 + 24 3 106 52 + 52 1 158 78 + 78 1
∗54 246 + 18 33 ∗108 492 + 36 66 ∗160 616 + 64 84
55 40 + 40 1 110 168 + 40 9
56 48 + 24 11 111 72 + 72 1
57 36 + 36 1 112 192 + 48 36
Table 1: Skew morphisms of cyclic groups of order n.
the asterisk character (∗). All included groups are listed by their orders, and for each of
them we provide the total number of skew morphisms (written as the sum of the numbers
of proper skew morphisms and automorphisms), and the number of conjugacy classes of
proper skew morphisms in Aut(Zn). Note that the automorphism group of a cyclic group
Zn is always abelian, and hence the conjugation action of Aut(Zn) on itself is trivial. It
follows that the number of conjugacy classes of Aut(Zn) is equal to |Aut(Zn)|. For this
reason, we list the number of conjugacy classes only for proper skew morphisms. We also
note that the numbers of skew morphisms in Table 1 for n ≤ 60 coincide with the numbers
of skew morphisms in [9].
5.3 Remarks concerning Table 1
An inspection of Table 1 suggests various interesting questions regarding skew morphisms
of cyclic groups. Possibly the most natural question to ask here is which values n actually
appear in Table 1 or, equivalently, which cyclic groups admit a proper skew morphism.
338 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
This was answered in [20] for cyclic groups, and later in [12] for all other abelian groups.
Specifically, if an abelian group A does not admit any proper skew morphism, then A is
cyclic of order n where n = 4 or gcd(n, ϕ(n)) = 1, or A is an elementary abelian 2-group.
Next, we look at values n such that Zn admits (up to conjugacy in Aut(Zn)) only one
proper skew morphism. In [20] this was shown to be true for all cases where n is a product
of two distinct primes and gcd(n, ϕ(n)) > 1. The only other value n that appears in Table 1
and has this property is n = 8. An interesting question raised in this context is whether this
covers all such values n, or if there are others.
We are also interested in those cyclic groups that admit only coset-preserving skew
morphisms. Unlike general skew morphisms, coset-preserving skew morphisms are well
understood for cyclic groups, and, in particular, we can list all coset-preserving skew mor-
phisms of Zn in polynomial time; see [6]. Thus, if for some n we can show that all skew
morphisms of Zn are coset-preserving, then we can find all skew morphisms of Zn much
faster than by using the algorithm explained in Section 5.2. In the following section we
completely solve this problem by characterising all cyclic groups that admit only coset-
preserving skew morphisms.
6 Cyclic groups that admit only coset-preserving skew morphisms
In this section we focus on cyclic groups admitting only coset-preserving skew morphisms.
Our main theorem is the following:
Theorem 6.1. All skew morphisms of Zn are coset-preserving if and only if n = 2em with
e ∈ {0, 1, 2, 3, 4} and m odd and square-free.
Note that Theorem 6.1 include all groups Zn that does not admit any proper skew
morphism, as in that case either n = 4, or (n, ϕ(n)) = 1, which forces n to be square-free
(for if some prime square p2 divides n, then p is a common factor of n and ϕ(n)). In what
follows, we will say that the positive integer n is resolvable if it is expressible in the form
n = 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. The proof of Theorem 6.1
is split into two parts; in Section 6.1 we show that if n is not resolvable, then Zn admits a
non-coset-preserving skew morphism, and in Section 6.2 we show that if n is resolvable,
then Zn does not admit a non-coset-preserving skew morphism. Then in Section 6.3 we use
Theorem 6.1 (and further facts about coset-preserving skew morphisms of cyclic groups) to
enumerate all skew morphisms for many finite cyclic groups for which no such enumeration
was available to date. Finally, in Section 6.4 we give an example that demonstrates how
Theorem 6.1 can be applied to find a precise formula for the number of skew morphisms
of Zn in the case when n is resolvable and has a relatively small number of prime factors.
6.1 Cyclic groups admitting non-coset-preserving skew morphisms
Here we show that if the order of a cyclic group is divisible by 32 or by the square of an
odd prime, then this group admits a skew morphism that does not preserve the cosets of its
kernel. To do this, we will use some facts about skew morphisms of Zn that give rise to a
regular Cayley map. First, we have the following:
Proposition 6.2 ([18]). A skew morphism φ of a finite group B gives rise to a regular
Cayley map for B if and only if the set of elements of some orbit of ⟨φ⟩ is closed under
taking inverses and generates B.
M. Bachratý: Quotients of skew morphisms of cyclic groups 339
We say that a skew morphism φ of a group B is t-balanced if its kernel has index 2
in B. The value t is given by t = π(a), where a is any element of B not contained in
kerφ. In the special case when t = ord(φ)− 1 we say that φ is anti-balanced. For further
information on t-balanced skew morphisms we refer the reader to [11]. The following
observation shows that every coset-preserving skew morphism of Zn that gives rise to a
regular Cayley map is either an automorphism of Zn, or a t-balanced skew morphism of
Zn.
Lemma 6.3. If φ is a coset-preserving skew morphism of Zn that gives rise to a regular
Cayley map, then the index of kerφ in Zn is at most two. In particular, if n is odd, then
kerφ = Zn and φ is an automorphism of Zn.
Proof. Since φ gives rise to a regular Cayley map, by Proposition 6.2 there exists some
orbit T of ⟨φ⟩ that is closed under taking inverses and generates Zn. Further, by [20,
Corollary 3.3] we know that T contains some element t such that ⟨t⟩ = Zn, and since
T = −T , we also have −t ∈ T . Next, from the fact that φ is coset-preserving we deduce
that t and −t are both in the same coset of kerφ in Zn. It follows that 2t ∈ kerφ, and
noting that t is a generator of Zn, we also have gcd(n, t) = 1. Hence 2 ∈ kerφ, and the
rest follows.
Throughout the proof of the following proposition we repeatedly refer to the classifica-
tion of regular Cayley maps for cyclic groups given in [13].
Proposition 6.4. If a positive integer n is divisible by 32 or p2 for some odd prime p, then
Zn admits a skew morphism that is not coset-preserving.
Proof. First, assume that n is odd and divisible by p2 for some odd prime p. Then there
exists a regular Cayley map for Zn with non-balanced representation (see [13, Section 8]),
and hence there exists a proper skew morphism φ of Zn that gives rise to this Cayley map.
Since φ is proper and n is odd, by Lemma 6.3 we deduce that φ is not coset-preserving.
Next, if n is even and divisible by p2 for some odd prime p, then we have Zn = Zℓ×Z2e
with ℓ odd. Since ℓ is clearly divisible by p2, from the previous paragraph we know that
Zℓ admits a skew morphism φ that is not coset-preserving. By Lemma 2.11 there exists a
skew morphism θ of Zn such that θ ↾Zℓ= φ and ker θ = kerφ×Z2e . Now it can be easily
seen that since φ does not preserve the cosets of kerφ in Zℓ, the same is true for θ and
cosets of ker θ in Zn.
Finally, let n be even and divisible by 32, and consider the factorisation Zn = Z2e ×Zℓ
with ℓ odd. Note that to show that Zn admits a non-coset-preserving skew morphism, it is
sufficient to prove this for Z2e (and the rest will follow by Lemma 2.11). Let M(2m, r)
be the regular Cayley map for Z2m given by [13, Definition 3.6]. This map is defined for
every unit r modulo m such that if b is the largest divisor of m that is relatively prime
to r − 1, then either b = 1, or r is a root of −1 modulo b of multiplicative order 2k
where k is relatively prime to m/b. Let m = 2e−1, r = 2e−3 + 1, and M = M(2m, r).
Note that the largest divisor of m relatively prime to r − 1 is 1, and hence b = 1. Also
note that r is not a root of −1 modulo m, and since e ≥ 5 we have r2 ̸≡ 1 (mod m).
It follows that M has no balanced, no t-balanced, and no anti-balanced representation;
see [13, Section 8]. Since every automorphism of Z2e gives rise to a skew morphism with
a balanced representation, and every skew morphism of Z2e with kernel of index 2 in Z2e
gives rise to a skew morphism with either t-balanced or anti-balanced representation, we
340 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
deduce that a skew morphism φ of Z2e that gives rise toM has kernel of index greater than
two in Z2e . Hence by Lemma 6.3 we find that φ is not coset-preserving.
6.2 Cyclic groups admitting only coset-preserving skew morphisms
Next we show that if the positive integer n is resolvable, then all skew morphisms of Zn
are coset-preserving. We start with the following technical lemma.
Lemma 6.5. Let φ be a skew morphism of a cyclic group Zn, let N be any non-trivial
subgroup of kerφ, let φ∗N be the skew morphism of Zn/N induced by φ, and let L/N be
the kernel of φ∗N . Also let s be a prime factor of n/|kerφ|, let ks denote the largest power
of s that divides |(kerφ)/N |, and let a = n/(s|kerφ|) be an element of Zn. If sks divides
|L/N |, then a /∈ kerφ and a ∈ L.
Proof. First, since a|kerφ| = n/s < n, we have a /∈ kerφ. (Note that this part is true
regardless of whether sks divides |L/N |.) Next, let K = kerφ, and let m be an integer
such that |K/N | = mks. (Observe that m and ks are relatively prime.) Then since K/N is
a subgroup of L/N , we know that mks divides |L/N |. But |L/N | is also divisible by sks,
and since gcd(m, ks) = 1, it follows that |L/N | must be divisible by msks. Hence |L| is
divisible by s|K|, and therefore a ∈ L.
We are now ready to prove the key part of the proof of Theorem 6.1.
Proposition 6.6. Let n = 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. Then
every skew morphism of Zn is coset-preserving.
Proof. Suppose to the contrary that the assertion is not true, and let n be the smallest
resolvable integer such that Zn admits a skew morphism φ that is not coset-preserving.
Also let K = kerφ, let φ∗ denote the skew morphism of Zn/K induced by φ, and let
φ be a quotient of φ. Since the only skew morphism of the trivial group is clearly coset-
preserving, we have n > 1. Recalling that every skew morphism of a non-trivial group has
a non-trivial kernel, it follows that |K| ≥ 2. In particular, it follows that |K| has at least
one prime factor. We proceed by considering the two following cases:
Case (a): |K| is a prime power
Let p be the largest prime divisor of n. Then by Corollary 3.3 we know that p di-
vides |K|. If p = 2, then n = 2, 4, 8, or 16, in which case Zn does not admit a skew
morphism that is not coset-preserving; see [9]. Hence p is odd and |K| = p, and thus
|Zn/K| = n/p. Then since p is the largest prime divisor of n, we know that p does not
divide |Zn/K|ϕ(|Zn/K|), and it follows from Theorem 2.9 that p does not divide the order
of φ∗. Further, by Lemma 3.1 we know that p divides ord(φ), and since ord(φ∗) = pφ,
we deduce that p divides ord(φpφ). On the other hand, noting that φpφ preserves the
cosets of K in Zn, we have ord(φpφ) ≤ |K| = p, and hence ord(φpφ) = p. Therefore
ord(φ) = ppφ, and then by Proposition 5.1(b) we have |kerφ| = p. But p does not di-
vide (n/p), which (by Lemma 4.1(a)) is the order of φ, and so by Lemma 3.1 we find that
φ is a group automorphism. Hence by Lemma 4.2 we deduce that φ is coset-preserving,
contradiction.
Case (b): |K| has at least two distinct prime factors
M. Bachratý: Quotients of skew morphisms of cyclic groups 341
Let k = |K|, and let d denote the integer satisfying n = kd. (Note that the elements of
K are exactly the multiples of d modulo n.) Also let d = r1 . . . rℓ be a factorisation such
that each factor is either an odd prime or the maximum possible power of two. Since φ does
not preserve the cosets of K in Zn, we know that φ(1) ̸≡ 1 (mod d). Hence, noting that
all factors ri for i ∈ {1, . . . , ℓ} are pairwise relatively prime, it follows from the Chinese
Remainder Theorem that there exists r ∈ {r1, . . . , rℓ} such that φ(1) ̸≡ 1 (mod r). We
will show that this cannot happen.
First, assume that r is odd. It follows that r is a prime, and also that n is not divisible
by r2. (And so, in particular, r does not divide |K|.) Let p be any prime factor of |K|, let
N be the unique subgroup of K of order p, and let φ∗N be the skew morphism of Zn/N
induced by φ. Also let L/N be the kernel of φ∗N , and suppose that |L/N | is not divisible
by r. Since the order of the cyclic group Zn/N is clearly resolvable, by the assumption of
minimality of n we know that φ∗N is coset-preserving. Then since r divides Zn/N but not
L/N , we have φ(1) ≡ φ∗N (1) ≡ 1 (mod r). But this contradicts the fact that φ(1) ̸≡ 1
(mod r), and hence we deduce that r divides |L/N |. Now, since r does not divide |K|, it
follows that r does not divide |K/N |, and thus we may use Lemma 6.5 (with s = r and
ks = 1). Hence we deduce that the element a = d/r of Zn is not contained in K, but
also a ∈ L, and by Lemma 2.6 it follows that pπ(a) ≡ p (mod ord(φ)). Since p was an
arbitrary prime factor of |K|, the same is true also for some other prime factor q of |K|.
(Here we use the assumption that the order of K has at least two distinct prime factors.)
Hence we have
pπ(a) ≡ p (mod ord(φ)),
qπ(a) ≡ q (mod ord(φ)),
for a pair of distinct primes p and q. Then since gcd(p, q) = 1, we deduce that π(a) ≡ 1
(mod ord(φ)), and consequently a ∈ K, contradiction.
Next, assume that r is even. Again let p denote a prime factor of |K|, and define N ,
φ∗N , and L/N same as in the case when r was odd. Also let k2 denote the largest power of
2 that divides |K/N |, and suppose that |L/N | is not divisible by 2k2. Noting that K/N is
a subgroup of L/N , it follows that the largest power of 2 that divides |L/N | must be k2.
Then since φ∗N must be coset-preserving (due to the minimality of n) and the largest power
of 2 that divides |Zn/N | is equal to rk2, we deduce that φ(1) ≡ φ∗N (1) ≡ 1 (mod r),
contradicting the fact that φ(1) ̸≡ 1 (mod r). Hence it follows that 2k2 divides |L/N |,
and Lemma 6.5 (in this case we take s = 2 and ks = k2) implies that the element a = d/2
of Zn satisfies a /∈ K and a ∈ L. Using the same argument as for r odd this again leads to
a contradiction.
Theorem 6.1 now follows directly from Propositions 6.4 and 6.6.
6.3 Enumeration
In [6] it was shown that each coset-preserving skew morphism φ of Zn is uniquely de-
termined by the following four parameters: the smallest non-zero element d of kerφ; the
element h of Zn such that φ(1) = 1+h; the smallest positive integer s such that φ(d) = sd;
and the positive integer e = π(1). (Note that s always exists since d ∈ kerφ and φ re-
stricts to an automorphism of kerφ.) Using various properties of coset-preserving skew
morphisms it can be checked that if φ is non-trivial, then the parameters d, h, s and e must
satisfy the following properties (see [6, Section 4] for details):
342 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
(i) all four parameters are positive integers;
(ii) d is a proper divisor of n;
(iii) s < n/d and gcd(s, n/d) = 1;
(iv) h is a multiple of d strictly smaller than n;
(v) if r is the smallest positive integer such that h
∑r−1
i=0 s
i ≡ 0 (mod n),
then e is a (multiplicative) unit modulo r of order d and e < r;
(vi) sd ≡
d−1∑
j=0
(1 + h
ℓj∑
i=0
si) (mod n), where ℓj = ej − 1 mod r; and
(vii) se−1 ≡ 1 (mod n/d).
On the other hand, for each set of parameters (d, h, s, e) satisfying all of the above prop-
erties there exists a unique non-trivial coset-preserving skew morphism of Zn (which can
be constructed in a straightforward way) with this parameter set; see [6, Section 5]. This
gives a one-to-one correspondence between non-trivial coset-preserving skew morphisms
of Zn and the sets of parameters (d, h, s, e), and this correspondence can be used to find
all coset-preserving skew morphisms of a given cyclic group in a polynomial time (in the
cardinality of the group). Hence, by Theorem 6.1 we can quickly find all skew morphisms
of Zn, where n is resolvable. Using the above observations, we developed an algorithm
that can enumerate all skew morphisms for any given cyclic group of any resolvable order.
A C++ implementation of this algorithm succeeded in enumerating all skew morphisms for
cyclic groups of resolvable orders smaller than 10000 within a second, even without paral-
lelisation. In comparison, the best available method to date for finding all skew morphisms
for cyclic groups of general order (described in Section 5.2) is computationally feasible
only up to order 161. In our enumeration, which is available at [1], we provide the total
number of skew morphisms of Zn, and also the total number of automorphisms and their
proportion among all skew morphisms.
6.4 Skew morphisms of Z4p
Although coset-preserving skew morphisms can be generated efficiently, there is no known
explicit formula for the number of coset-preserving skew morphisms of Zn for general n.
Such a formula would be useful, and in the case when n is resolvable it would give the
number of all skew morphisms of Zn. Resolvable integers n with the simplest structure
(with respect to their prime factorisation) are primes, in which case all skew morphisms of
Zn are automorphisms of Zn. The situation is also completely understood in the case when
n is a product of two distinct primes p and q, in which case the number of skew morphisms
of Zpq is (p−1)(q−1) if gcd(p, q−1) = gcd(p−1, q) = 1, and 2(p−1)(q−1) otherwise;
see [20] for example. Here we go one step further and find a formula for the number of
skew morphisms of Z4p, where p is an odd prime. We will use the following fact:
Proposition 6.7. Let p be an odd prime. If φ is a proper skew morphism of Z4p, then the
action of φ on its kernel is trivial.
M. Bachratý: Quotients of skew morphisms of cyclic groups 343
Proof. Let π and K denote the power function and the kernel of φ, and let φ∗ be the skew
morphism of Zn/K induced by φ. Note that by Theorem 6.1 we know that φ is coset-
preserving, and so φ∗ must be trivial. Further, by Corollary 3.3 we know that p divides
|K|, and since φ is proper it follows that the order of K is either p or 2p. We proceed by
considering these two cases. In each case, we let T be the orbit of ⟨φ⟩ that contains 1. Note
that by Proposition 2.7 we have |T | = ord(φ).
Case (a): |K| = p
Since φ is coset-preserving, we know that the cosets of K in Zn are preserved set-wise
by φ. Note that only one element of the coset 1 +K does not generate Zn (either p or 3p,
depending on whether p ≡ 1 (mod 4) or p ≡ 3 (mod 4)), and since by Lemma 2.12 no
elements outside of K are fixed by φ, it follows that each orbit of ⟨φ⟩ on 1 +K generates
Zn. Hence by Proposition 2.7 we know that all of these orbits have size ord(φ), and since
|1 +K| = |K| = p, we see that ord(φ) = p. Since φ restricts to an automorphism of K,
and ord(φ) = |K| = p, we deduce that the action of φ on K is trivial.
Case (b): |K| = 2p
In this case, since φ is proper, by Theorem 2.9 we have gcd(ord(φ), 4p) > 1. If the
order of φ is odd, then we have ord(φ) = p or ord(φ) = 3p, but the latter case can be
easily excluded since φ must preserve both cosets of K in Zn of size 2p.
Next we deal with the case when ord(φ) is even. Note that by Lemma 4.1(a) we have
π(1) = φ(1), and since φ is an automorphism of the cyclic group Zord(φ) of even order,
we deduce that π(1) is odd. We will use this observation to show that both p and 3p are
contained in some orbits of ⟨φ⟩ that generate Zn. Suppose to the contrary that this is not
true. Since every element of 1 + K other than p and 3p generates Zn and no element of
1 +K is fixed by φ, we must have φ(p) = 3p and φ(3p) = p. Then since π(1) is odd, we
have φπ(1)(p) = 3p, and therefore φ(1+p) = φ(1)+φπ(1)(p) = φ(1)+3p. On the other
hand, we have φ(p+ 1) = φ(p) + φπ(p)(1) = 3p+ φπ(p)(1). But then φ(1) = φπ(p)(1),
and since |T | = ord(φ) we find that π(p) = 1. This forces p ∈ K, contradicting the fact
that the order of K is 2p. Hence we deduce that all orbits of ⟨φ⟩ on 1 +K generate Zn. In
particular, ord(φ) divides 2p, and it follows that ord(φ) = 2p.
We have shown that if |K| = 2p, then ord(φ) is equal to p or 2p. Hence by order
considerations it can be easily seen that φ acts on K either trivially, or by the inversion. To
exclude the latter, first note that 1 and φ(1) are in the same coset of K in Z4p, and hence
φ(1)− 1 ∈ K. If we let h = φ(1)− 1, then we have φ2(1) = φ(h+1) = φ(h) +φ(1) =
−h + h + 1 = 1, but then by Proposition 2.7 we have ord(φ) = 2, which is impossible.
Hence we again conclude that the action of φ on K is trivial.
Using Theorem 6.1 and Proposition 6.7 we can now easily enumerate all proper skew
morphisms of Z4p.
Theorem 6.8. If p is an odd prime, then the number of skew morphisms of Z4p is{
6p− 6 if p ≡ 1 (mod 4)
4p− 4 if p ≡ 3 (mod 4).
Proof. Throughout this proof we refer to the properties (i) to (vii) and the parameters d,
h, s and e of coset-preserving skew morphisms for cyclic groups explained in Section 6.3.
344 Ars Math. Contemp. 24 (2024) #P2.08 / 327–346
We know that |Aut(Z4p)| = 2p − 2, so we proceed by counting proper skew morphisms
of Z4p. Let φ be a proper skew morphism of Z4p, and recall that by Theorem 6.1 it is
coset-preserving. Let d, h, s and e be the four defining parameters of φ, and note that by
Proposition 6.7 we have s = 1. Since p is the largest prime divisor of 4p, by Corollary 3.3
we know that p divides |kerφ|, and it follows that d = 2 or d = 4. (The case d = 1 can be
excluded as φ is proper.)
First let d = 2, and let h be any positive multiple of 2 strictly smaller than 4p. If 4
divides h, then by (v) we find that r = p and e = p− 1. Since s = 1, both (iii) and (vii) are
trivially true, and (vi) holds as (1+h)+(1+h(p−1)) ≡ 2+hp ≡ 2 (mod 4p). If 4 does not
divide h and h ̸= 2p, then by (v) we have r = 2p and e = 2p− 1. Again both (iii) and (vii)
hold trivially, and (vi) is also true since (1+h)+(1+h(2p−1) ≡ 2+2hp ≡ 2 (mod 4p).
If h = 2p, then it can be easily verified that ord(φ) = r = 2, contradicting the fact that
φ is proper. Since for all but one choice of h we obtain exactly one coset-preserving skew
morphism, it follows that in this case we have exactly 2p− 2 skew morphism of Z4p.
Next let d = 4, and let h be any positive multiple of 4 strictly smaller than 4p. Then by
(v) we deduce that r = p, and e must be a fourth root of unity modulo p. It follows that
necessarily p ≡ 1 (mod 4), in which case there are two possible candidates for e. Note
that for either candidate we have e2 ≡ −1 (mod p) and e3 ≡ −e (mod p). Again (iii)
and (vii) hold trivially, and (vi) holds as well since (1 + h) + (1 + he) + (1 + h(p− 1)) +
(1 + h(p − e)) ≡ 4 + 2hp ≡ 4 (mod 4p). Hence, if p ≡ 1 (mod 4), then every choice
of h gives two coset-preserving skew morphisms (one for each choice of e), which gives a
total of 2p− 2 skew morphisms of Z4p.
ORCID iDs
Martin Bachratý https://orcid.org/0000-0002-4300-7507
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.09 / 347–354
https://doi.org/10.26493/1855-3974.2751.81f
(Also available at http://amc-journal.eu)
Finite simple groups on triple systems*
Xiaoqin Zhan , Xuan Pang , Suyun Ding †
School of Science, East China JiaoTong University,
Nanchang, 330013, People’s Republic of China
Received 3 December 2021, accepted 14 May 2023, published online 22 November 2023
Abstract
Let D be a triple system, and let G be a finite simple group. In this paper we almost
determine all possibilities of D admitting G as its flag-transitive automorphism group.
Keywords: Triple system, flag-transitivity, finite simple group.
Math. Subj. Class. (2020): 05B07, 20B25, 05B25
1 Introduction
A 2-(v, k, λ) design is a pair D = (P,B) where P is a set of v points and B is a collection
of b k-subsets (blocks) of P with the property that every 2-subset of P occurs in λ blocks
of B. If no blocks are identical, then D is called simple.
An automorphism of a design D is a permutation of P which leaves B invariant. The
full automorphism group of D, denoted by Aut(D), is the group consisting of all automor-
phisms of D. A flag of D is a point-block pair (α,B) such that α ∈ B. For G ≤ Aut(D),
G or D is called flag-transitive if G acts transitively on the set of flags, and point-primitive
if G acts primitively on P . A set of blocks of D is called a set of base blocks with respect
to an automorphism group G of D if it contains exactly one block from each G-orbit on the
block set. In particular, if G is a flag-transitive automorphism group of D, then any block
B is a base block of D.
*The authors would like to express their gratitude to the referee who made very helpful comments and sugges-
tions that improved our paper. This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12361004 and 11961026) and the Natural Science Foundation of Jiangxi Province (Grant Nos.
20224BAB211005 and 20224BAB201005).
†Corresponding author.
E-mail addresses: zhanxiaoqinshuai@126.com (Xiaoqin Zhan), p1443202623@163.com (Xuan Pang),
dingsy2017@163.com (Suyun Ding)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
348 Ars Math. Contemp. 24 (2024) #P2.09 / 347–354
In this paper, we focus on simple 2-(v, 3, λ) designs also known as simple triple sys-
tems, which can be denoted by TS(v, λ). One possibility is to take all possible 3-subsets
of P however such designs are called complete and will be ignored. A triple system is a
Steiner triple system, or STS(v), when λ = 1.
Let r be the number of the blocks through a given point. For a TS(v, λ), it is well
known that a necessary and sufficient condition for the existence of a TS(v, λ) is v ̸= 2
and λ ≡ 0 (mod (v − 2, 6)), and
3b = vr; (1.1)
r =
λ(v − 1)
2
; (1.2)
b =
λv(v − 1)
6
; (1.3)
b ≥ v. (1.4)
A 2-(v, k, 1) design is also called a finite linear space. A classic result is that of Higman
and McLaughlin [8] who proved that for a finite linear space, flag-transitivity implies point-
primitivity. Then Buekenhout, Delandtsheer and Doyen in [1] proved that if G acts flag-
transitively on a linear space, then G is of affine or almost simple type. In 1990, the
six-person team [2] classified all flag-transitive linear spaces apart from those with an one-
dimensional affine automorphism group.
For 2-(v, k, 1) designs with small values of k, one of the first classifications was for
Steiner triple systems in [4], which considered what happens when the action was block-
transitive but not 2-transitive on points. It is described in [11] what happens when the
action on points is 2-transitive. This result depends on the classification of all finite simple
groups and is subsumed into the general results proved by Kantor in [10].
Let G be a flag-transitive automorphism group of a TS(v, λ). It is shown in [6, 2.3.7(c),
(e)] that G is point-primitive. Moreover, we can easily prove that G is 2-homogeneous (see
Lemma 2.2 below). This result makes it possible to classify all flag-transitive triple systems
using the classification of the finite 2-transitive permutation groups. Our main purpose is
to give a classification of all triple systems admitting a simple flag-transitive automorphism
group.
We now state the main result of this paper:
Theorem 1.1. Let D be a triple system, and let G be a finite simple group. If G acts
flag-transitively on D, then one of the following LINES of Table 1 holds.
Remark 1.2.
• All but the triple systems listed in LINES 20 and 21 exist.
• If G = PSU(3, q) with q = 5, then there are only two flag-transitive triple systems
corresponding to LINES 19 and 20.
• The existence of triple systems with 3 ∤ q and q ̸= 5 corresponding to LINES 20 and
21 is in doubt.
X. Zhan et al.: Finite simple groups on triple systems 349
Table 1: G and corresponding triple systems.
LINE G D Notes
1 A7 TS(15, 1)
2 TS(15, 12)
3 PSL(2, 11) TS(11, 3)
4 TS(11, 6)
5 HS TS(176, 12)
6 TS(176, 72)
7 TS(176, 90)
8 Co3 TS(276, 112)
9 TS(276, 162)
10 PSp(2d, 2) TS(2d−1(2d + 1), 22d−2) d ≥ 3
11 TS(2d−1(2d + 1), 2(2d−1 − 1)(2d−2 + 1))
12 PSp(2d, 2) TS(2d−1(2d − 1), 22d−2) d ≥ 3
13 TS(2d−1(2d − 1), 2(2d−1 + 1)(2d−2 − 1))
14 PSL(d, q) TS( q
d−1
q−1 , q − 1) d ≥ 3
15 TS( q
d−1
q−1 ,
qd−1
q−1 − q − 1)
16 PSL(2, q) TS(q + 1, q−1
2
) q ≡ 1(mod 4)
17 Ree(q) TS(q3 + 1, 2(q − 1)) q = 32e+1 > 3
18 TS(q3 + 1, q − 1)
19 PSU(3, q) TS(q3 + 1, q − 1) q ≥ 3
20 TS(q3 + 1, q
2−1
(3,q+1)
)
21 TS(q3 + 1, 2(q
2−1)
(3,q+1)
)
2 Useful lemmas
The notation and terminology used is standard and can be found in [5, 6] for design theory
and in [7, 9] for group theory. In particular, if G is a permutation group on a set Ω, and
{α, β} ⊆ ∆ ⊆ Ω, then Gα denotes the stabilizer of a point α in G, and Gαβ denotes the
pointwise stabilizer of two points α and β in G, and G∆ denotes the setwise stabilizer of
∆ in G.
The following result about flag-transitive 2-designs is well-known.
Lemma 2.1. Let D = (P,B) be a 2-(v, k, λ) design, and let G be an automorphism group
of D. For any α ∈ P and B ∈ B, G is flag-transitive if and only if G is point-transitive
and Gα is transitive on the pencil P (α) (the set of blocks through α), if and only if G is
block-transitive and GB is transitive on the points of B.
Lemma 2.2. Let D = (P,B) be a triple system, and let G be a flag-transitive automor-
phism group of D. If G is a simple group, then G acts 2-transitively on P .
Proof. Let {α, β} and {γ, δ} be arbitrary two unordered pairs of P . By the definition of a
triple system, there are two points ε and θ such that B1 = {α, β, ε} and B2 = {γ, δ, θ} are
two blocks of D. The flag-transitivity of G implies that there is a g ∈ G such that
(ε,B1)
g = (εg, Bg1 ) = (θ,B2),
and so {α, β}g = {γ, δ}. Thus G is 2-homogeneous. If G is a simple group, then G acts
2-transitively on P by [7, Theorem 9.4B].
350 Ars Math. Contemp. 24 (2024) #P2.09 / 347–354
Lemma 2.3. Let D = (P,B) be a triple system, and let G ≤ Aut(D) be a 2-transitive
group on P . Then the following conditions are equivalent:
(i) G acts flag-transitively on D.
(ii) If B = {α, β, γ} ∈ B, then {{α, β, γi} | γi ∈ γG{α,β}} is the set of all blocks
through points α and β.
Proof. (i) ⇒ (ii): Let B(α, β) = {B1, B2, . . . , Bλ} be the set of blocks through points α
and β, where Bi = {α, β, γi}, γi ∈ P \ {α, β}. Clearly, B(α, β)G{α,β} = B(α, β). If G
acts flag-transitively on D, then for any two flags (γi, Bi) and (γj , Bj), there is a g ∈ G
such that (γi, Bi)g = (γj , Bj), so γ
g
i = γj and g ∈ G{α,β}. Thus G{α,β} acts transitively
on B(α, β) and hence {γ1, . . . , γλ} = γ
G{α,β}
i .
(ii) ⇒ (i): Let (γ,B) and (ϵ, C) be two flags of D with B = {α, β, γ}, C = {δ, η, ϵ}.
By the 2-transitivity of G, there exists g1 ∈ G such that {α, β}g1 = {δ, η}, thus Bg1 =
{δ, η, γg1} is a block containing δ and η. Since {{δ, η, ϵi} | ϵi ∈ ϵG{δ,η}} is the set
of all blocks through δ and η, there exists g2 ∈ G{δ,η} such that γg1g2 = ϵ, and then
(γ,B)g1g2 = (ϵ, C). Therefore, G acts flag-transitively on D.
Corollary 2.4. Let G be a 2-transitive group on a point set P with |P| = v, and let
λ1, λ2, . . . , λk be all sizes of orbits of Gαβ on P \ {α, β}. If λi ̸= λj for i ̸= j, then there
exist k different flag-transitive TS(v, λi).
Proof. Without loss of generality, let ∆ = γGαβ with |∆| = λ1, where γ ∈ P \ {α, β}.
Since Gαβ ⊴ G{α,β}, the group Gαβ acts 12 -transitively on γ
G{α,β} , that is, Gαβ-orbits
on γG{α,β} have the same length. The uniqueness of the Gαβ-orbit with size λ1 implies
that γGαβ = γG{α,β} . Thus G{α,β} has a unique orbit with size λ1. Let B = {α, β, γ}
and B = BG. We shall prove below that D = (P,B) is a TS(v, λ1) admitting G as its
flag-transitive automorphism group.
Since G is 2-transitive, for any pair {δ, η}, there exists g ∈ G such that {α, β}g =
{δ, η}. So G{δ,η} has a unique orbit ∆g = (γg)G{δ,η} with |∆g| = |∆| = λ1. Let
B(δ, η) be the set of elements of B containing δ, η with |B(δ, η)| = λ. It is easy to see that
Λ = {{δ, η, ϵ} | ϵ ∈ ∆g} ⊆ B, so we have λ ≥ λ1. On the other hand, for C = {δ, η, θ} ∈
B(δ, η), there exists h ∈ G such that C = Bh. As |γGαβ | = |αGγβ | = |βGαγ | = λ1,
we may assume that θ = γh. Then |θG{δ,η} | = |γhG{δ,η} | = |γG{α,β}h| = |∆h| = λ1,
it implies λ1 ≥ λ. Thus, λ = λ1 and B(δ, η) = Λ. Hence D is a TS(v, λ1), and G is a
flag-transitive automorphism group of D by Lemma 2.3(ii).
Lemma 2.5. Let G be a 2-transitive group on a point set P with |P| = v, and let ∆ =
{α, β, γ} be a 3-subset of P . If Gαβ is a cyclic group of order λ and |γGαβ | = λ, then
(i) D = (P,∆G) is a flag-transitive TS(v, λ) if and only if G∆∆ ∼= S3, or
(ii) D = (P,∆G) is a flag-transitive TS(v, 2λ) if and only if G∆∆ ∼= Z3.
Proof. Here we only prove case (i), and case (ii) can be proved by same procedure. Since
Gαβ is a cyclic group for any points α and β, we have that G∆ = G∆∆. Let D = (P,∆G).
If D is a flag-transitive TS(v, λ), then using Lemma 2.1 and Equation (1.3), we have that
b =
λv(v − 1)
6
= |∆G| = [G : G∆] = [G : Gαβ ][Gαβ : G∆].
X. Zhan et al.: Finite simple groups on triple systems 351
By 2-transitivity of G and |Gαβ | = λ, we obtain |G∆| = 6. The flag-transitivity of G
implies that G∆ acts transitively on the points of ∆ by Lemma 2.1. Thus G∆ ∼= S3.
If G∆ ∼= S3, then G{α,β}γ ∼= Z2 and |∆G| = [G : Gαβ ][Gαβ : G∆] = λv(v−1)6 . Thus,
D is a TS(v, λ) as G acts 2-transitively on P . Clearly,
|γG{α,β} | = [G{α,β} : G{α,β}γ ] = λ,
where G{α,β}γ = G{α,β} ∩Gγ . Therefore, G acts flag-transitively on D by Corollary 2.4.
Lemma 2.6. Let G = Ree(q) act 2-transitively on Ω, where |Ω| = q3+1 and q = 32e+1 >
3. Then there exist subsets ∆, Σ of size 3 such that
G∆∆ = Z3, GΣΣ = S3.
Proof. Let Q be a Sylow 3-subgroup of G. Then |Q| = q3, and there exists α ∈ Ω such that
Q is regular on Ω\{α}. Thus each subgroup of Q is semiregular on Ω\{α}. Let x, y ∈ Q
such |x| = |y| = 3, x /∈ Z(Q) and y ∈ Z(Q), where the centre Z(Q) is elementary abelian
of order q.
Let ∆ be an orbit of ⟨x⟩. Then |∆| = 3 and G∆∆ = Z3 or S3. Further, since x is not
conjugate to x−1 in G (reference [12]), we have G∆∆ ∼= ⟨x⟩ ∼= Z3.
Consider y acting on Ω \ {α}. Since y is in the centre Z(Q), there is an involution
z ∈ Gα such that yz = y−1, and the subgroup H = ⟨y, z⟩ ∼= S3. Since ⟨y⟩ is semiregular
on Ω \ {α}, the set Ω \ {α} is divided into 13q
3 orbits of ⟨y⟩:
∆1,∆2, . . . ,∆m,
where m = 13q
3 is odd. Since each H-orbit Σ contains a ⟨y⟩-orbit, the cardinality |Σ| = 3
or 6. As the number 13q
3 of ⟨y⟩-orbits is odd, it follows that there is at least one H-orbit Σ
on Ω \ {α} has length 3. Therefore, GΣΣ = HΣΣ = S3 with |Σ| = 3.
3 Proof of Theorem 1.1
Let D = (P,B) be a TS(v, λ), and let G be a simple group acting flag-transitively on D.
Then G acts 2-transitively on P by Lemma 2.2. Since we neglect the case D is complete,
we may assume that G is not 3-homogeneous group on P . Thus, all such groups are known
and we can find a classification in [3] and we have that G must be one of the following
Table 2.
We will prove Theorem 1.1 by analyzing the 11 cases in Table 2 one by one.
Proof of Theorem 1.1. Let α and β be two points of P . For Cases 1 – 7, we have the
following facts by the proof of [10, Theorem 1]:
If G = A7 and v = 15, then Gαβ has orbit-lengths 1 and 12 on P \ {α, β}.
If G = PSL(2, 11) and v = 11, then Gαβ has orbit-lengths 3 and 6 on P \ {α, β}.
If G = HS and v = 176, then Gαβ has orbit-lengths 12, 72 and 90 on P \ {α, β}.
If G = Co3 and v = 276, then Gαβ has orbit-lengths 112 and 162 on P \ {α, β}.
If G = PSp(2d, 2) and v = 22d−1 + 2d−1, then Gαβ has orbit-lengths 2(2d−1 −
1)(2d−2 + 1) and 22d−2 on P \ {α, β}.
352 Ars Math. Contemp. 24 (2024) #P2.09 / 347–354
Table 2: 2-transitive, not 3-homogeneous simple groups.
Case Group Degree Notes
1 A7 15
2 PSL(2, 11) 11
3 HS 176
4 Co3 276
5 PSp(2d, 2) 22d−1 + 2d−1 d ≥ 3
6 PSp(2d, 2) 22d−1 − 2d−1 d ≥ 3
7 PSL(d, q) (qd − 1)/(q − 1) d ≥ 3
8 PSL(2, q) q + 1 q ≡ 1 (mod 4)
9 Suz(q) q2 + 1 q = 22e+1 > 2
10 Ree(q) q3 + 1 q = 32e+1 > 3
11 PSU(3, q) q3 + 1 q ≥ 3
If G = PSp(2d, 2) and v = 22d−1 − 2d−1, then Gαβ has orbit-lengths 2(2d−1 +
1)(2d−2 − 1) and 22d−2 on P \ {α, β}.
If G = PSL(d, q) with d ≥ 3 and v = q
d−1
q−1 , Gαβ has orbit-lengths q − 1 and
qd−1
q−1 − q − 1 on P \ {α, β}.
It follows from Corollary 2.4 that D is one of triple systems corresponding LINES 1-15
in Table 1.
Case 8: G = PSL(2, q) with q ≡ 1 (mod 4) and v = q + 1. In this case, there are
exactly two G-orbits on 3-subsets of q + 1 points with size q(q
2−1)
12 . Also, Gαβ
∼= Z q−1
2
has two orbits with length q−12 on P \ {α, β}, denoted by Γ1 and Γ2. Suppose that Γ1 =
{α1, α2, . . . , α q−1
2
}, Γ2 = {β1, β2, . . . , β q−1
2
}. For i ∈ {1, 2}, let Di = (P,∆Gi ) where
∆i = {α, β, γi} and γi ∈ Γi. It is easy to calculate that |G∆i | = 6, and hence G∆i ∼= S3.
By Lemma 2.5(i), both D1 and D2 are TS(q + 1, q−12 ). Let
g = (α, β)(α1, β1) · · · (α q−1
2
, β q−1
2
).
Clearly, g is an isomorphism from D1 to D2, that is D1 ∼= D2. Thus, D is a TS(q+1, q−12 ).
Case 9: G = Sz(q) and v = q2 + 1. Since G acts flag-transitively on D, then 3 | |G| by
Lemma 2.1. But this contracts the fact that 3 ∤ |G| (see [9, Theorem 3.6]). Therefore, there
is no triple system admitting Sz(q) as its flag-transitive automorphism group.
Case 10: G = Ree(q) and v = q3 +1 with q = 32e+1 > 3. From Lemmas 2.5 and 2.6, we
have that D is one of triple systems corresponding LINES 17 and 18 in Table 1.
Case 11: G = PSU(3, q) and v = q3 + 1. Since Gαβ ∼= Z q2−1
(3,q+1)
has a unique orbit O
with size q − 1 and q(3, q + 1) orbits with size q
2−1
(3,q+1) . Similar to proof of Lemma 2.4,
we can prove that there exists a unique TS(q3 +1, q− 1) admitting G as its flag-transitive
automorphism group.
If q = 3e ≥ 3, there exist subsets ∆, Σ of size 3 such that G∆∆ = Z3, GΣΣ = S3 by the
same proof as Lemma 2.6. In this case, D is one of triple systems corresponding LINES 20
and 21 in Table 1 from Lemma 2.5.
X. Zhan et al.: Finite simple groups on triple systems 353
If q = 5 then D can only be a flag-transitive TS(126, 8) in addition to TS(126, 4) by a
simple calculation. This means that there is no flag-transitive TS(126, 16) in this case.
Unfortunately, we don’t know whether Lemma 2.6 holds when 3 ∤ q. Thus the existence
of TS(q3 + 1, q
2−1
(3,q+1) ) (or TS(q
3 + 1, 2(q
2−1)
(3,q+1) )) with 3 ∤ q and q ̸= 5 is in doubt.
This completes the proof of Theorem 1.1.
Conjecture 3.1. Let D be a triple system TS(q3 + 1, λ), and let G = PSU(3, q) act
flag-transitively on D with 3 ∤ q and q ̸= 5. If λ ̸= q − 1 then one of following holds:
(i) If q is even, then λ = 2(q
2−1)
(3,q+1) .
(ii) If q is odd, then λ = q
2−1
(3,q+1) or
2(q2−1)
(3,q+1) .
In fact, using MAGMA, we have already proved that the conjecture holds when q ≤ 100.
ORCID iDs
Xiaoqin Zhan https://orcid.org/0000-0003-0669-6419
Xuan Pang https://orcid.org/0000-0003-2500-9741
Suyun Ding https://orcid.org/0000-0002-6564-4427
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.10 / 355–383
https://doi.org/10.26493/1855-3974.2763.1e6
(Also available at http://amc-journal.eu)
There is a unique crossing-minimal rectilinear
drawing of K18*
Bernardo M. Ábrego , Silvia Fernández–Merchant
Departament of Mathematics, California State University at Northridge,
CA, United States
Oswin Aichholzer
Institute for Software Technology, University of Technology, Graz, Austria
Jesús Leaños †
Academic Unit of Mathematics, Autonomous University of Zacatecas, Mexico
Gelasio Salazar
Institute of Physics, Autonomous University of San Luis Potosi, Mexico
Received 8 December 2021, accepted 15 June 2023, published online 14 February 2024
Abstract
We show that, up to order type isomorphism, there is a unique crossing-minimal recti-
linear drawing of K18. It is easily verified that this drawing does not contain any crossing-
minimal drawing of K17. Therefore this settles, in the negative, the following question
from Aichholzer and Krasser: is it true that, for every integer n ≥ 4, there exists a crossing-
minimal drawing of Kn that contains a crossing-minimal drawing of Kn−1?
Keywords: Rectilinear crossing number, complete graphs, k-edges.
Math. Subj. Class. (2020): 05C10, 05C60
*We thank an anonymous referee for carefully reading an earlier version of this paper, and providing several
insightful comments, corrections, and suggestions.
†Corresponding author.
E-mail addresses: bernardo.abrego@csun.edu (Bernardo M. Ábrego), silvia.fernandez@csun.edu (Silvia
Fernández–Merchant), oaich@ist.tugraz.at (Oswin Aichholzer), jleanos@uaz.edu.mx (Jesús Leaños),
gsalazar@ifisica.uaslp.mx (Gelasio Salazar)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
356 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
1 Introduction
The rectilinear crossing number cr(G) of a graphG is the minimum number of edge cross-
ings in a rectilinear drawing of G in the plane, i.e., a drawing of G in the plane where the
vertices are points in general position and the edges are straight line segments. A drawing
of G with exactly cr(G) crossings is crossing-minimal.
Determining the rectilinear crossing number cr(Kn) of the complete graph Kn is a
well-known open problem in combinatorial geometry (see for instance [5, 11]). In [9]
Aichholzer et al. determined the exact values of cr(Kn) for 13 ≤ n ≤ 17. In that paper
also the following question was raised.
Question 1.1. Is it true that, for every integer n ≥ 4, there exists a crossing-minimal
drawing of Kn that contains a crossing-minimal drawing of Kn−1?
The exact value of cr(Kn) is known for n ≤ 27 and n = 30 (see [3, 7, 8, 9, 10]). The
value of cr(K18) = 1029 was established in [8]. Crossing-minimal rectilinear drawings of
Kn for this range of values of n can be found in [2] and [6]. In particular, from [6], we
know that there are at least 37269 non-isomorphic crossing-minimal drawings of K17.
Let θ denote the counterclockwise rotation of 2π/3 around the origin, and let
W := {(−51, 113), (6, 834), (16, 989), (18, 644), (18, 1068), (22, 211)}. From [2], we
know that the 18-point set W ∪ θ(W ) ∪ θ2(W ) induces a crossing-minimal drawing of
K18. See Figure 1 for an illustration of such a point set.
Our main result is the following.
Theorem 1.2. Up to order type isomorphism, there is a unique 18-point set whose induced
rectilinear drawing of K18 has cr(K18) crossings.
Let D be the (unique, in view of Theorem 1.2) crossing-minimal rectilinear drawing
of K18. It is easily verified that every subdrawing of D with 17 points has more than
cr(K17) = 798 crossings. This settles Question 1.1 in the negative.
In the next section, we introduce the necessary notation and additional concepts re-
quired for the proof of Theorem 1.2. In Section 4 we prove Theorem 1.2.
2 k-edges, (≤ k)-edges, and 3-decomposability
Throughout this section, Q is a set of n ≥ 3 points in general position in the plane. If p
and q are distinct points of Q, then we denote by pq the directed line spanned by p and q,
directed from p towards q. Furthermore, pq+ and pq− denote the set of points in Q on the
right and left, respectively, of pq. Thus Q = pq− ∪ {p, q} ∪ pq+ for all p, q ∈ Q with
p ̸= q.
Let k ∈ {0, 1, . . . , ⌊n/2⌋−1}. A k-edge ofQ is a directed line spanned by two distinct
points of Q, which leaves exactly k points of Q on one side. A (≤ k)-edge (respectively,
a (> k)-edge) is an i-edge of Q with 0 ≤ i ≤ k (respectively, k < i ≤ ⌊n/2⌋ − 1).
Let Ek(Q), E≤k(Q), and E>k(Q) denote, respectively, the set of k-edges, (≤ k)-edges
and (> k)-edges of Q. We use ek(Q), e≤k(Q), and e>k(Q) to denote, respectively, the
number of elements in Ek(Q), E≤k(Q), and E>k(Q). Then e≤k(Q) =
∑k
j=0 ej(Q) and
e>k(Q) =
(
n
2
)
− e≤k(Q).
The vector E≤k(Q) := (e≤0(Q), e≤1(Q), . . . , e≤⌊n/2⌋−1(Q)) is the (≤ k)-edges vec-
tor of Q. Finally, e≤k(n) denotes the minimum of e≤k(P ) taken over all n-point sets P
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 357
Figure 1: This is the 18-point set produced by the union of W = {(−51, 113), (6, 834),
(16, 989), (18, 644), (18, 1068), (22, 211)}, θ(W ) and θ2(W ). It is not difficult to see that
P produces a crossing-minimal rectilinear drawing ofK18. The triangle and the six straight
line segments show that P is 3-decomposable.
in the plane in general position. The exact determination of e≤k(n) is another well known
open problem in combinatorial geometry (see for instance [3, 4, 7, 8]).
The number of crossings in a rectilinear drawing of Kn and the number of k- and
(≤ k)-edges in its underlying n-point set P are closely related by the following equality,
independently proved in [4] and [12]:
cr(P ) =
⌊n/2⌋−2∑
k=0
(n− 2k − 3) e≤k (P )−
3
4
(
n
3
)
+
(
1 + (−1)n+1
) 1
8
(
n
2
)
. (2.1)
This equality allows us to fully determine the (≤ k)-edges vector of any 18-point set
whose induced drawing attains the rectilinear crossing number of K18.
Proposition 2.1. If P is an 18-point set such that cr(P ) = cr(K18), then E≤k(P ) =
(3, 9, 18, 30, 45, 63, 87, 120, 153).
Proof. Let Q be an 18-point set in the plane in general position. It is known (see [3]
or [7]) that E≤k(Q) = (e≤0(Q), e≤1(Q), . . . , e≤8(Q)) is bounded below entry-wise by
(3, 9, 18, 30, 45, 63, 87, 120, 153). On the other hand, from (2.1) we know that
cr(Q) = −612 + 15 · e≤0 (Q) + 13 · e≤1 (Q) + · · ·+ 1 · e≤7 (Q) .
From the coefficients of this equation and the fact that e≤8 (Q) = 153, it follows that if
e≤k (Q) is greater than the k-th component in the vector (3, 9, 18, 30, 45, 63, 87, 120, 153),
then cr(Q) > 1029.
358 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Finally, we introduce a concept that captures a property shared by all known crossing-
minimal rectilinear drawings ofKn, for n a multiple of 3. A point setQ is 3-decomposable
if it can be partitioned into three equal-size sets A,B and C, such that (i) there exists a
triangle T enclosing the point set Q; and (ii) the orthogonal projection of Q onto the three
sides of T shows A between B and C on one side, B between C and A on the second
side, and C between A and B on the third side. In such a case, we say that {A,B,C}
is a 3-decomposition of Q. For instance, {W, θ(W ), θ2(W )} is a 3-decomposition of the
18-point set shown in Figure 1.
As in [2], if {A,B,C} is a 3-decomposition of Q, we define two types of edges. Let
p and q be distinct points of Q. If p, q ∈ A, p, q ∈ B or p, q ∈ C then we call pq
monochromatic; otherwise, pq is bichromatic. Let Emonk (Q) and E
bi
k(Q) denote the set of
monochromatic and bichromatic k-edges of Q, respectively. As before, we use emonk (Q) and
ebik(Q) to denote |Emonk (Q)| and |Ebik(Q)|, respectively. Note that ek(Q) = emonk (Q)+ ebik(Q).
Now we partition the monochromatic edges of Q into three types. If p, q ∈ A, then we
say that pq is an edge of type aa. Similarly, we define the edges of types bb and cc. For
x ∈ {a, b, c}, we denote the number of monochromatic k-edges of type xx by exxk (Q).
Then emonk (Q) = e
aa
k (Q) + e
bb
k (Q) + e
cc
k (Q).
3 Overview of the proof of Theorem 1.2
For the rest of this paper, P is an 18-point set in the plane in general position where the
rectilinear crossing number of K18 is attained. That is, cr(P ) = cr(K18).
The first step in the proof, carried out in Section 4.1, consists of giving an algo-
rithm that yields a canonical, unambiguous labelling of the points in P . The 18 points
in P get labelled x0, x1, . . . , x5, y0, y1, . . . , y5, z0, z1, . . . , z5. Thus P gets naturally par-
titioned into three sets X = {x0, x1, x2, x3, x4, x5}, Y = {y0, y1, y2, y3, y4, y5}, and
Z := {z0, z1, z2, z3, z4, z5}. As we shall prove shortly afterwards, {X,Y, Z} happens to
be a 3-decomposition of P .
Once we have laid out the foundation by giving a canonical labelling of the points of
P , the rest of the proof consists of showing the following:
Lemma 3.1 (Implies Theorem 1.2). For each pair of distinct points p, q ∈ P , the set pq+
is uniquely determined.
Clearly Lemma 3.1 implies Theorem 1.2: if the lemma holds, then the unambiguity of
the labelling of the points in P implies that P is unique up to order type isomorphism.
First we establish the lemma for the case in which pq is a (≤5)-edge. This is actually
done in Section 4.1, where we give the algorithm to label the points in P . Indeed, the
unambiguity in the labelling of the points in P is established in Proposition 4.3(1), and in
order to prove this we need to prove simultaneously Proposition 4.3(2), which in particular
implies Lemma 3.1 for the case in which pq is a (≤5)-edge.
We then move on to proving Lemma 3.1 for the case in which pq is a (>5)-edge, that is,
when pq is either a 6-edge, or a 7-edge, or an 8-edge. As we shall see, even if this follows
from elementary observations, the investigation of these cases is remarkably more involved
than the case in which pq is a (≤5)-edge.
The first step towards the investigation of (>5)-edges is given in Section 4.2, where we
prove that {X,Y, Z} is a 3-decomposition of P . This allows us to classify each edge of
P as either monochromatic or bichromatic, as we explained at the end of Section 2. Also
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 359
in Section 4.2 we show that for each k ∈ {6, 7, 8} it is easy to determine the number of
bichromatic k-edges and the number of monochromatic k-edges.
After proving these elementary properties of P we move on to Section 4.3. This is the
most technical and long part of the paper, and its purpose is to establish a collection of
structural properties of P . On a first read it may be advisable to skip this section, and only
come back to it whenever its main results are invoked in Sections 4.4 and 4.5.
Finally, in Section 4.4 (respectively, Section 4.5) we prove Lemma 3.1 for the case in
which pq is a monochromatic (respectively, bichromatic) 6-edge, 7-edge, or 8-edge. As we
shall see, using the structural results from Section 4.3 these tasks are reduced to a relatively
straightforward case analysis.
For completeness, the conclusion of the proof is presented in Section 4.6.
4 Proof of Theorem 1.2
We recall that throughout this paper, P is an 18-point set in the plane in general position
such that cr(P ) = cr(K18).
4.1 The algorithm to label the 18 points in P , and proof of Lemma 3.1 when pq is a
(≤ 5)-edge
It follows from Proposition 2.1 that the convex hull of P has exactly 3 vertices. Without
loss of generality (the whole set P may be rotated, if necessary) we may assume that all
three vertices have distinct x-coordinates. Let x0 denote the vertex with the smallest x-
coordinate. As we travel counterclockwise along the convex hull starting from x0, let y0
be the first vertex we find, and let z0 be the other vertex. See Figure 2.
Figure 2: The convex hull of P .
360 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Observation 4.1. E0(P ) = {x0y0, y0z0, x0z0}. □
We have already unambiguously determined a labelling for the three convex hull ver-
tices of P . It remains to unambiguosly determine a labelling for the remaining 15 points of
P .
For j ∈ {0, . . . , 5}, let x↷j denote the j-th point in P that we find as we rotate the line
y0x0 clockwise around y0 (we consider x0 to be the 0-th point in P hit by the rotating line,
so that x↷0 = x0). We define y
↷
j and z
↷
j similarly, using z0y0 and x0z0 as the clockwise
rotating lines, around z0 and x0, respectively. See Figure 3(a).
In an analogous manner, we let x↶j denote the j-th point in P that we find as we rotate
the line z0x0 counterclockwise around z0 (again, we consider x0 to be the 0-th point in
P hit by the rotating line, so that x↶0 = x0). We define y
↶
j and z
↶
j similarly, using
x0y0 and y0z0 as the counterclockwise rotating lines, around x0 and y0, respectively. See
Figure 3(b).
Figure 3: (a) As we rotate the line y0x0 clockwise around y0, the third point in P we find
is labelled x↷3 . The points x
↷
1 and x
↷
2 are also indicated. (b) As we rotate the line z0x0
counterclockwise around z0, the third point in P we find is labelled x↶3 . The points x
↶
1 and
x↶2 are also indicated. By definition x
↷
0 = x
↶
0 = x0. Note that in this example x
↷
i = x
↶
i
for i = 0, 1, 2, 3.
Observation 4.2. For each j ∈ {0, . . . , 5}, y0x↷j , z0x↶j , z0y↷j , x0y↶j , x0z↷j , and y0z↶j
are all j-edges.
The next statement is our first major result on the structure of P . In particular, it yields
a labelling of all the points of P . As it happens, this proposition simultaneously establishes
Lemma 3.1 for the case in which pq is a (≤ 5)-edge.
Proposition 4.3. Let j ∈ {0, . . . , 5}. Then:
(1) For u ∈ {x, y, z}, u↷j and u↶j are the same point, which will be denoted uj;
(2) For all nonnegative integers m,n such that m+ n = j, we have that
(a) Ej(P ) = {umvn
∣∣m+n = j and uv ∈ {xy, yz, zx}}. Moreover, for such values
of m,n, and j the following holds:
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 361
(b) umv+n = {ui
∣∣ i < m} ∪ {vi∣∣ i < n} for any uv ∈ {xy, yz, zx}.
Proof. We prove (1) and (2) by induction on j. Since x↷0 = x
↶
0 = x0, y
↷
0 = y
↶
0 = y0,
and z↷0 = z
↶
0 = z0, it follows from Observations 4.1 and 4.2 that (1) and (2) are true for
j = 0. Now we let t ∈ {0, 1, 2, 3, 4} be an integer such that (1) and (2) hold for every j
such that 0 ≤ j ≤ t (in particular, the points xj , yj , zj are already defined for 0 ≤ j ≤ t).
We complete the proof by showing that then (1) and (2) hold for j = t+ 1.
Let Xt := {x0, . . . , xt}, Yt := {y0, . . . , yt}, Zt := {z0, . . . , zt}, and Pt := Xt ∪ Yt ∪
Zt. From the definitions involved, it follows that |Pt| = 3(t + 1). First we establish an
injection ψ : Pt → Et+1(P ).
Consider any point xi ∈ Xt. It follows from the induction hypothesis that xiyt−i is a
t-edge, and that xiy+t−i = {xr
∣∣ r < i} ∪ {yr ∣∣ r < t− i}.
Let xi be the first point that we find as we rotate the line xiyt−i counterclockwise
around xi. It is easy to see that the induction hypothesis implies that the rotating line hits
xi with its head, and so xix+i = {xr
∣∣ r < i}∪{yr ∣∣ r < t−i+1}. We define ψ(xi) = xixi.
In an analogous manner we define ψ(yi) and ψ(zi) for all yi ∈ Yt and zi ∈ Zt. Since ψ
defines a one-to-one relation and |Pt| = 3(t+ 1), it follows that |ψ(Pt)| = 3(t+ 1).
Let E′ := {x0z↷t+1, y0x↷t+1, z0y↷t+1}. Observation 4.2 implies that E′ ⊂ Et+1(P ).
We note that ψ(x0) = x0y↶t+1, ψ(y0) = y0z
↶
t+1, and ψ(z0) = z0x
↶
t+1. Using these
observations and that {x↷t+1, y↷t+1, z↷t+1} ∩ Pt = ∅, it follows that E′ ∩ ψ(Pt) = ∅. On the
other hand, from Proposition 2.1 it follows that |Et+1(P )| = 3(t + 2). Thus Et+1(P ) is
the disjoint union of ψ(Pt) and E′.
By way of contradiction, suppose that x↷t+1 ̸= x↶t+1. Then each point of {x1, . . . , xt}
is contained in the interior of the quadrilateral bounded by y0x↷t+1, z0x
↶
t+1, z0x0, and x0y0
(see Figure 4). From the induction hypothesis it follows that x↷t+1x
↶
t+1 /∈ E≤t(P ). This
and the fact that x↷t+1x
↶
t+1 /∈ ψ(Pt)∪E′ imply that |x↷t+1x↶t+1−| ≥ t+2. Then the interior
of the triangle T bounded by y0x↷t+1, z0x
↶
t+1, and x
↷
t+1x
↶
t+1 is nonempty, and, moreover,
it contains every element of Q := x↷t+1x
↶
t+1
− \ {x0, x1, . . . , xt}. Let p be the first point
of Q that z0x↶t+1 finds as it is rotated counterclockwise around x
↶
t+1. Then x
↶
t+1p must be
a (t + 1)-edge of P . On the other hand, it is immediately seen that x↶t+1p /∈ ψ(Pt) ∪ E′,
contradicting that Et+1 = ψ(Pt) ∪ E′.
This contradiction shows that x↷t+1 and x
↶
t+1 are the same point. Analogous arguments
show that y↷t+1 and y
↶
t+1 are the same point, and that z
↷
t+1 and z
↶
t+1 are the same point. This
proves (1) for j = t+ 1.
Now we show that (2) holds for j = t + 1. Note that at this point xt+1, yt+1, zt+1
are all well-defined. For each m ∈ {0, 1, . . . , t + 1}, we let Xm := {xi
∣∣ i ≤ m},
Ym := {yi
∣∣ i ≤ m}, and Zm := {zi ∣∣ i ≤ m}.
Let m ∈ {0, 1, 2, . . . , t+ 1}. We shall show that
(i) xmyt+1−m is in Et+1(P ); and
(ii) xmy+t+1−m = Xm−1 ∪ Yt−m.
By symmetry, analogous arguments show that: (i’) ymzt+1−m is in Et+1(P );
(ii’) ymz+t+1−m = Ym−1 ∪ Zt−m; (i”) zmxt+1−m is in Et+1(P ); and (ii”) zmx
+
t+1−m =
Zm−1 ∪ Xt−m. Note that these six assertions, together with the fact that |Et+1(P )| =
3(t+ 2), imply (2).
Thus we complete the proof by showing (i) and (ii).
362 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Figure 4: If x↷t+1 ̸= x↶t+1, then the triangle T contains a point p such that x↶t+1p is a (t+1)-
edge of P .
Since xt+1 = x↶t+1 = x
↷
t+1, yt+1 = y
↶
t+1 = y
↷
t+1, and zt+1 = z
↶
t+1 = z
↷
t+1, it follows
that (i) and (ii) hold whenever m is in {0, t + 1}. Thus it suffices to prove (i) and (ii) for
1 ≤ m ≤ t.
From the induction hypothesis we have that xm−1y+t+1−m = Xm−2 ∪ Yt−m and
xmy
+
t−m = Xm−1 ∪ Yt−m−1. Also note that Xm−1 ∪ Yt−m ⊆ xmy+t+1−m.
LetB denote the triangle bounded by the lines xm−1yt+1−m, xmyt−m, and xmyt+1−m
(see Figure 5). Let PB be the set of points of P contained in the interior of B.
We claim that PB = ∅. By way of contradiction, suppose this is not the case. Let
L = p1p2 · · · pk be the lower chain of the convex hull of PB ∪ {xm, yt+1−m}. Then
p1 = xm and pk = yt+1−m, where (since B ̸= ∅) k ≥ 3. We note that pip+i+1 =
Xm−1 ∪Yt−m for all i = 1, 2, . . . , k− 1. Thus each edge of L is a (t+1)-edge. We recall
that Et+1(P ) = ψ(Pt)∪E′. It is readily seen that no edge of L is in E′, and so every edge
of L is in ψ(Pt). In particular, the line p2p3 is in ψ(Pt).
Recall that every edge in ψ(Pt) is obtained by starting with a line viwt−i (for v, w ∈
{x, y, z}, v ̸= w), counterclockwise rotating it around vi, and recording the first point
p in P hit by the rotating line: ψ(vi) is then the line vip. Thus, in particular p2p3 is
obtained in this way. Now if we reverse the process and clockwise rotate p2p3 around
p2, the first point hit by the rotating line must be yt−m. This implies that p2 = xm,
contradicting that p1 = xm. We therefore conclude that PB = ∅. Finally, note that
PB = ∅ immediately implies that ψ(xm) = xmyt+1−m. Thus xmyt+1−m is a (t + 1)-
edge. This proves (i). Moreover, as we observed above, Xm−1 ∪ Yt−m ⊆ xmy+t+1−m.
Since |Xm−1 ∪ Yt−m| = t+ 1, then Xm−1 ∪ Yt−m = xmy+t+1−m. Thus (ii) follows.
In view of Proposition 4.3(1), we have achieved our goal to unambiguously identify
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 363
Figure 5: If the interior of the triangle B is nonempty, then every edge of the convex chain
p1, p2, . . . , pk is a (t+ 1)-edge.
364 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
(and label) all 18 points of P . For the rest of the paper, we letX := {x0, x1, . . . , x5}, Y :=
{y0, y1, . . . , y5}, and Z := {z0, z1, . . . , z5}, where xj , yj , and zj are as in Proposition 4.3,
for j = 0, 1, . . . , 5.
4.2 {X,Y, Z} is a 3-decomposition of P
If we rotate the line x0z0 clockwise along x0, then for j = 1, 2, . . . , 5, the j-th point hit
by the rotating line is zj . If we rotate the line x0y0 counterclockwise along x0, then for
j = 1, 2, . . . , 5, the j-th point hit by the rotating line is yj . It follows that the sixth point hit
by the clockwise rotating line ℓx is in X , and the sixth point hit by the counterclockwise
rotation line ℓ′x is also in X (see Figure 6). These two points in X are obviously distinct
(since |X| > 2), and so they define an infinite cone CX with vertex x0 (here by cone with
vertex p we mean a pair of distinct directed rays, both with startpoint p). Note that CX is
the smallest infinite cone with vertex x0 that contains X . See Figure 6.
We similarly find infinite cones CY (with vertex y0) and CZ (with vertex z0).
Figure 6: The sets X,Y, and Z are contained in the indicated (closed) shaded regions. The
shaded region containing X is ∆X,Y ∩∆X,Z .
Now CX ∪ CY divide the plane into several regions, three of which are bounded. Two
of these bounded regions are triangles: one triangle ∆X,Y with x0 as a vertex and another
triangle ∆Y,X with y0 as a vertex; the other one is a quadrilateral. The entire set X is
contained in ∆X,Y , and the entire set Y is contained in ∆Y,X . By considering the pair
CX , CZ (respectively, CY , CZ), we obtain triangles ∆X,Z and ∆Z,X (respectively, ∆Y,Z
and ∆Z,Y ). Thus X ⊆ ∆X,Y ∩ ∆X,Z , Y ⊆ ∆Y,X ∩ ∆Y,Z , and Z ⊆ ∆Z,X ∩ ∆Z,Y .
Hence the situation is as illustrated in Figure 6. In this figure, each of ∆X,Y ∩ ∆X,Z ,
∆Y,X ∩∆Y,Z , and ∆Z,X ∩∆Z,Y is a quadrilateral, although it is easy to see that any (or
all) of them may be a triangle.
In view of this, it follows immediately that there is a triangle that witnesses the follow-
ing:
Proposition 4.4. P is 3-decomposable, with 3-decomposition {X,Y, Z}. □
As we mentioned in Section 2, knowing that {X,Y, Z} is a 3-decomposition of P
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 365
allows us to classify each edge of P as either monochromatic or bichromatic: for p, q ∈ P ,
the edge pq is monochromatic if p and q belong to the same set of the 3-decomposition
{X,Y, Z}. Otherwise, pq is bichromatic.
We close this section by noting that using the 3-decomposability of P it is easy to
determine the number of bichromatic and monochromatic k-edges in P , for each k ∈
{0, . . . , 8}.
Indeed, since P is 3-decomposable it follows from [2, Claim 1] that ebi≤k(P ) = 3
(
k+2
2
)
for k ∈ {0, . . . , 5}, ebi≤6(P ) = 81, and ebi≤7(P ) = 99. Also note that ebi≤8(P ) is the
total number of bichromatic edges of P , namely 3 · 6 · 6 = 108. Using that ebij(P ) =
ebi≤j(P )− ebi≤j−1(P ) for j ∈ {1, . . . , 8}, we obtain the following.
Proposition 4.5. ebik (P ) = 3(k + 1) for k ∈ {0, . . . , 5}, ebi6 (P ) = 18, ebi7 (P ) = 18, and
ebi8 (P ) = 9.
To obtain emon6 (P ), e
mon
7 (P ) and e
mon
8 (P ), we note that Proposition 2.1 implies that
e6(P ) = 24, e7(P ) = 33, and e8(P ) = 33. Since ej(P ) = ebij (P ) + e
mon
j (P ) for
j = 0, . . . , 8, Proposition 4.5 implies the following.
Corollary 4.6. emonk (P ) = 0 for k ∈ {0, . . . , 5}, emon6 (P ) = 6, emon7 (P ) = 15, and
emon8 (P ) = 24.
4.3 Structural properties of P
4.3.1 Determination of euuk (P ) for any u ∈ {x, y, z} and any k ∈ {0, . . . , 8}
Let u ∈ {x, y, z}, i ∈ {1, 2, . . . , 5}, and ℓu, ℓ′u be the directed rays forming the cone
CU with vertex u0 mentioned in the arguments leading to Proposition 4.4. See Figure 6.
From now on, we shall use ui0 to denote the i-th point of P that ℓu finds when it is rotated
clockwise around of u0 until it reaches ℓ′u. Clearly, {u10, u20, . . . , u50} = {u1, u2, . . . , u5}.
Our next observation is evident, but useful.
Observation 4.7. Let v1, v2, and v3 be three distinct points in P , and let ℓ := v1v2 and
ℓ′ := v1v3. Let P1 := ℓ− ∩ ℓ′+, P2 := ℓ+ ∩ ℓ′+, and P3 := ℓ− ∩ ℓ′−. Then P1, P2, and
P3 are pairwise disjoint subsets of P . See Figure 7. For i = 1, 2, 3, let ri be the number of
points in Pi. If P \ {v1, v2, v3} is the disjoint union of P1, P2, and P3, and pi is the i-th
point of P1 that ℓ finds when it is rotated counterclockwise around v1 until it reaches ℓ′,
then v1pi is a j-edge of P for j = min{r2 + i, 16− (r2 + i)}.
The next observation is immediate from the definition of ui0 and Observation 4.7.
Observation 4.8. Let u ∈ {x, y, z}. Then
(1) u0u10 and u0u
5
0 are both 6-edges,
(2) u0u20 and u0u
4
0 are 7-edges and they are the only 7-edges of the type u0u, and
(3) u0u30 is an 8-edge.
Claim 4 in [2] implies that euu8 (P ) ≤ 8 for each u ∈ {x, y, z}. Using this, together
with Observation 4.8 and Corollary 4.6, we obtain the following.
Proposition 4.9. Let u ∈ {x, y, z}. Then euuk (P ) = 0 for k ∈ {0, . . . , 5}, euu6 (P ) = 2,
euu7 (P ) = 5, and e
uu
8 (P ) = 8.
366 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Figure 7: The i-th point of P1 := ℓ− ∩ ℓ′+ that ℓ finds when it is rotated counterclockwise
around v1 is a j-edge for j = min{r2 + i, 16− (r2 + i)}, where r2 denotes the number of
points of P2 := ℓ+ ∩ ℓ′+.
The next corollary follows immediately from Observation 4.8(1) and Proposition 4.9.
Corollary 4.10. Emon6 (P ) = {x0x10, x0x50, y0y10 , y0y50 , z0z10 , z0z50}. Moreover, any other
monochromatic edge must belong to Emon7 (P ) ∪ Emon8 (P ).
4.3.2 Determination of the convex hull of U for U ∈ {X,Y, Z} and related facts
Let u be any element of {x, y, z}. One of the main goals in this subsection is to show
that the triangle formed by u0, u4 and u5 contains in its interior the remaining u’s, namely,
u1, u2 and u3. We also prove other statements about the relative position of the elements
of U . Almost all these assertions will be used in the subsequent steps later on.
Proposition 4.11. Let u ∈ {x, y, z}. If u1 ∈ {u20, u40}, then there are at least three 7-edges
of type uu involving u1 but not u0.
Proof. We prove the proposition for the case u = x. The cases u = y and u = z are
handled in a totally analogous manner.
Suppose that x1 = x40. Then ℓ := x0x1 leaves x
1
0, x
2
0 and x
3
0 on its left halfplane
(and x50 on its right halfplane). Let z
′ be the first z that ℓ finds when it is rotated counter-
clockwise around x1, and let ℓ′ = x1z′. See Figure 8. By Corollary 4.10, we know that
{x1x10, x1x20, x1x30} ⊂ Emon7 (P ) ∪ Emon8 (P ). Then ℓ finds each of x10, x20 and x30 before
it reaches z′. This and Observation 4.7 imply that at most one of x1x10, x1x
2
0, x1x
3
0 is an
8-edge and, by Corollary 4.10, the other two must be 7-edges.
From the way in that the x′s were labelled it follows that x50 is the first x that ℓ finds
when it is rotated clockwise around x1, and so x1x50 is a (≤ 7)-edge. This and Corol-
lary 4.10 imply that x1x50 is the third required 7-edge. The case x1 = x
2
0 can be handled in
an analogous manner (y′s play the role of z’s).
Proposition 4.12. Let u ∈ {x, y, z} and {p, q} = {x, y, z} \ {u}. Suppose that
{q0, . . . , q5} ⊂ u0u3−0 and that {p0, . . . , p5} ⊂ u0u
3+
0 . Then:
(A1) u1 /∈ {u10, u50};
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 367
Figure 8: Here x0x1 is a 7-edge.
(A2) there are at least two 7-edges of type uu involving u1 but not u0;
(A3) u2 /∈ {u10, u50};
(A4) each of u3u4, u3u5 and u4u5 is an 8-edge;
(A5) {u10, u50} = {u4, u5};
(A6) the triangle formed by u0, u4 and u5 is the convex hull of U ; and
(A7) if u5 ∈ u0u+4 , then u0u
−
4 = {q0, . . . , q5} and u0u
+
5 = {p0, . . . , p5}. Otherwise,
u0u
+
4 = {p0, . . . , p5} and u0u
−
5 = {q0, . . . , q5} .
Proof. By rotating P if necessary, and exchanging appropriately the labels x, y, and z, we
can assume, without any loss of generality, that u = x, p = y, q = z and that X,Y and Z
are placed as in Figure 9.
(A1): Seeking a contradiction, suppose that x10 = x1. Let v be the first point that x0x1
finds when it is rotated clockwise around x1 as shown in Figure 9(a). Note that v ∈ Y ,
as otherwise v ∈ {x2, x3, x4, x5} and x1v− = Z. Then x1v is a 6-edge, contradicting
Corollary 4.10. Let x′ be the last element of {x2, x3, x4, x5} that x1v finds when it is
rotated clockwise around x1. Since v ∈ Y , then x1x′ must be a (≤ 6)-edge, contradicting
Proposition 4.9. The case x50 = x1 can be handled in an analogous manner (with the roles
of Z and Y interchanged).
(A2): From (A1) we know that x0x1 leaves at least one x in each side. By definition
of x1, the points x2, x3, x4 and x5 must be contained in X ′ := X ∩ x1z+0 ∩ x1y
−
0 , see
Figure 9(b). Let x′ be the last element of {x2, x3, x4, x5} that x0x1 finds when it is rotated
clockwise around x1 as shown in Figure 9(b). Note that x1x′ must be a (≤ 7)-edge. Since
x′ ̸= x0, then Proposition 4.9 implies that x1x′ must be a 7-edge. Similarly, if we rotate
x0x1 in the other direction, then we can find the other 7-edge involving x1 but not x0.
(A3): Seeking a contradiction, suppose that x2 = x10. Then x0x2 is a 6-edge, and
x3, x4 and x5 are contained in X ′′ := X ∩x10y−0 . See Figure 10(b). From Observation 4.7,
we know that at most one of x2x3, x2x4, x2x5 is an 8-edge. This and Corollary 4.10 imply
368 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Figure 9: (a) x0x1 cannot be a 6-edge. (b) There are at least two 7-edges of type xx
involving x1 but not x0.
that at least two of x2x3, x2x4, x2x5 are 7-edges. This together with Observation 4.8(2)
and (A2) imply that exx7 (P ) ≥ 6, which contradicts Proposition 4.9. The case x2 = x50 can
be handled in an analogous manner.
(A4): From Corollary 4.10, Observation 4.8(1), and (A1), we know that x0x1 is a 7-
edge or an 8-edge. First suppose that x0x1 is a 7-edge. Then x1 ∈ {x20, x40}. This together
with Propositions 4.9 and 4.11 and Observation 4.8(2) imply that each element of Exx7 (P )
contains at least one of x0 or x1. This fact and Proposition 4.9 imply that x3x4, x3x5 and
x4x5 are 8-edges, as required.
Now, we suppose that x0x1 is an 8-edge. Then x2 ∈ x0x−1 or x2 ∈ x0x
+
1 . We
only analyze the case x2 ∈ x0x−1 (the other case is symmetric). Then we must have
that X ′ := X ∩ x0x+1 ∩ x2y
−
0 contains exactly two elements x
′, x′′ of {x3, x4, x5}, see
Figure 10(a). Now we rotate x2y0 clockwise around x2 until it be parallel to x0x1. See
Figure 10(a). From Observation 4.7 and Corollary 4.10, we know that at least one of
x2x
′, x2x
′′ is a 7-edge. Such a 7-edge plus the four 7-edges provided by Observation 4.8(2)
and (A2) give us, 5, the total number of 7-edges of P . This and Proposition 4.9 imply that
x3x4, x3x5, x4x5 are 8-edges, as required.
(A5): Seeking a contradiction, suppose that {x10, x50} ̸= {x4, x5}. Then (A1) and (A3)
imply that x3 = x10 or x3 = x
5
0. Again, by symmetry it is enough to analyze the case
x3 = x
1
0. Clearly, both x1 and x2 are contained in the triangle formed by x0x3, x0x
5
0 and
x3y0; and x4, x5 are contained in X ′′ := X ∩ x10y−0 , see Figure 10(b).
Now we rotate x0x3 clockwise around x3 until it reaches x3y0, and note that such a
rotation hits x4 and x5. From Observation 4.7 we know that at most one of x3x4 or x3x5
is 8-edge, contradicting (A4).
(A6): This follow directly from (A5) and the way in that the x’s were labelled.
(A7): Suppose that x5 ∈ x0x+4 . Then (A5) implies that x10 = x4 and x50 = x5. The
required equalities follow from the definition of x10 and x
5
0 and the hypotheses Z ⊂ x0x3−0
and Y ⊂ x0x3+0 . Similarly, we can deduce that x0x
−
5 = Z and x0x
+
4 = Y whenever
x5 ∈ x0x−4 .
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 369
Figure 10: (a) x3x4, x3x5, x4x5 are 8-edges. The dotted straight line containing x2 is
parellel to x0x1. (b) {x10, x50} = {x4, x5}, and hence x0x4 and x0x5 are 6-edges.
4.3.3 Determination of the position of u5 with respect to u0u4 for each u ∈{x, y, z}
Our main goal in this subsection is to show that u5 ∈ u0u+4 for each u ∈ {x, y, z}. In order
to prove this, we need to establish some auxiliary statements that will also be used later on.
Let u, v ∈ {x, y, z} with u ̸= v. We will say that u4u5 splits the v’s to mean that u4u5
separates {u0, . . . , u3} ∪ {v0, . . . , v3} from the rest of the points of P \ {u4, u5}.
Proposition 4.13. Let {u, v, w} = {x, y, z}, and suppose that u0u5 separates u4 from the
v’s. Then u4u5 splits the v’s.
Proof. By rotating and/or reflecting P along u0u4, if necessary, we can assume that u =
x, v = y, w = z and that X,Y and Z are placed as in Figure 6.
Since x0x5 separates x4 from the y’s, then x5 ∈ x0x+4 . From Proposition 4.3 we know
that x4y+0 = {x0, x1, x2, x3}. Then (A6) implies that x4y
+
0 ⊆ x4x
+
5 . If we rotate x4y0
counterclockwise around x4 until it reaches x0x4, then x5 ∈ x0x+4 , (A6), and Observa-
tion 4.7 together imply that at most one element of {x4x5} ∪ {x4y
∣∣y ∈ Y } is an 8-edge.
This and (A4) imply that such an 8-edge must be x4x5. Then (A6) implies that x4x5 leaves
exactly four y’s on its right. Moreover, from Proposition 4.3 it is easy to see that y0 and y1
are in x4x+5 . Let yi and yj be two elements of Y in x4x
−
5 . Without loss of generality, we
can assume that i < j. Then 2 ≤ i < j ≤ 5.
From (A6) we know that the triangle formed by y0, y4, and y5 is the convex hull of Y .
This implies that j ∈ {4, 5}. Seeking a contradiction, suppose that i ∈ {2, 3}.
Let T be the triangle formed by x0x5, x0yi and x4x5. See Figure 11(b). By the way y’s
were labelled, we know that if yr ∈ T then r ∈ {i+1, . . . , 5}\{j}. Let y′ be the first point
in Y ∩ T that yix0 finds when it is rotated clockwise around yi. See Figure 11(b). Then
yiy
′ is a (≤ i + 4)-edge because the points of P lying in the left side of yiy′ is a subset
of {x0, . . . , x3, y0, . . . , yi−1}. If i = 2, then yiy′ is a (≤ 6)-edge which does not involve
y0, contradicting Corollary 4.10. Finally, if i = 3, then yiy′ is a (≤ 7)-edge, contradicting
(A4).
Observation 4.14. From Proposition 4.13 we know that if {u, v, w} = {x, y, z}, then
u4u5 splits the v’s or the w’s.
370 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Figure 11: (a) If x4 ∈ x0x−5 , then x4x5 splits Y . (b) There are exactly two y’s, namely yi
and yj , in x4x−5 .
Proposition 4.15. Let u and v be two distinct elements of {x, y, z}. If u4u5 splits the v’s,
and v3v5 leaves u0 and v2 on the same side, then v4v5 splits the u’s.
Proof. By rotating P if necessary and exchanging appropriately the labels x, y and z, we
can assume that u = x and that X,Y and Z are placed as in Figure 6.
CASE 1: Suppose that x4x5 splits the y’s. Then we need to show that if y3y5 leaves x0
and y2 on the same side, then y4y5 splits the x’s.
From (A4), we know that y4y5 is an 8-edge, and from (A6), that y4y5 is in the convex
hull of Y .
First, we show that y5 ∈ y0y−4 . By way of contradiction, suppose this is not the case.
Then y5 ∈ y0y+4 and the triangle formed by y0, y4 and y5 looks like in Figure 12(a). Since
x4x5 splits the y’s, then x4x5 separates y3 from y4 and y5, and so all the x’s are on the
left side of both y3y4 and y3y5. In particular, x0 ∈ y3y−5 , and hence y2 ∈ y3y
−
5 . Then
y2 ∈ y3y−5 ∩ z0y
−
3 , and so y2 is contained in the triangle Q formed by y0y4, z0y3, y3y5.
See Figure 12(b). Since y3y4 is an 8-edge by (A4), and y3y4 leaves {y0, y2, x0, . . . , x5}
on its left, then it leaves y1, y5 on its right. This and the fact that y1 ∈ z0y−3 imply that
y1 is contained in the triangle R formed by z0y3, y3y4, y0y5. See Figure 12(b). Then y2y4
and y0y1 are 7-edges. This, together with Observation 4.8(2) and Proposition 4.11, implies
eyy7 (P ) ≥ 6, the required contradiction. Thus, we can conclude that y0y4 leaves the x’s
and y5 on the same side, and the desired result follows from Proposition 4.13.
CASE 2: x4x5 splits the z’s. Follow the same argument as in CASE 1 with left, right,
y, z,− and + in place of right, left, z, y,+ and −, respectively.
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 371
Figure 12: (a) y5 ∈ y0y+4 . (b) y2 is in Q and y1 is in R.
Proposition 4.16. If u ∈ {x, y, z}, then u3u5 leaves u0 and u1 on the same side.
Proof. As in Proposition 4.15, we can assume that u = x and thatX,Y andZ are placed as
in Figure 6. From (A4), we know that x3x5 is an 8-edge. Seeking a contradiction, suppose
that x3x5 separates x0 from x1.
First, suppose that x4x5 splits the y’s. Since P is placed as in Figure 6, then x0 ∈ x3x+5 ,
and hence, x1 ∈ x3x−5 . Then x3x
+
5 = {x0, x2}∪Y , or equivalently, x3x
−
5 = {x1, x4}∪Z.
This fact has two immediate consequences. The first one is that x1 = x20. This fact and
Proposition 4.11 imply that there are at least three 7-edges of type xx involving x1 but
not x0. The second consequence is that x2 is in the triangle X ′ (see Figure 13) formed
by x1y0, x3x5 and x0x5, and hence x2x5 must be a 7-edge too. The existence of these
four 7-edges together with those in Observation 4.8(2) imply exx7 (P ) ≥ 6, contradicting
Proposition 4.9.
Now suppose that x4x5 splits the z’s. Again, since P is placed as in Figure 6, then x0 ∈
x3x
−
5 and x1 ∈ x3x
+
5 . By similar arguments as above, we can deduce that x1 = x
4
0 and that
x2x5 is a 7-edge. As before, x1 = x40 and Proposition 4.11 imply that there are at least three
7-edges of type xx involving x1 but not x0. The existence of these four 7-edges together
with those in Observation 4.8(2) imply exx7 (P ) ≥ 6, contradicting Proposition 4.9.
Proposition 4.17. There is a u ∈ {x, y, z} such that u3u5 separates u4 from the other u’s.
Proof. From Proposition 4.16 we know that u3u5 leaves u0 and u1 on the same side for
each u ∈ {x, y, z}. Seeking a contradiction, we suppose that u3u5 separates {u0, u1} from
{u2, u4} for each u ∈ {x, y, z}.
By rotating and/or reflecting P if necessary, and exchanging appropriately the labels
x, y and z, we can assume that X,Y and Z are placed as in Figure 11(a), and that x4x5
splits the y’s. An immediate consequence of these assumptions and our hypothesis is that
(i) x3x5 leaves to z0 and x2 on its left side.
372 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
Figure 13: Here x1 ∈ x3x−5 .
Now suppose that y5 ∈ y0y+4 . Then y0y5 separates y4 from the z’s, and from Proposi-
tion 4.13 we know that y4y5 splits the z’s. In particular, the triangle formed by y0, y4 and
y5 must be as in Figure 12(a) and y0 ∈ y3y+5 . By supposition, y3y5 separates y0 from y2,
and so y3y5 leaves to x0 and y2 on its left side. This last and Proposition 4.15 imply that
y4y5 splits the x’s too, which is impossible. Thus we conclude that y5 ∈ y0y−4 . This and
Proposition 4.13 imply that y4y5 splits the x’s. This fact and our supposition imply that (ii)
y3y5 leaves to z0 and y2 on its right side.
If z4z5 splits the x’s, then (i) and Proposition 4.15 imply that x4x5 splits the z’s, which
contradicts that x4x5 splits the y’s. Similarly, if z4z5 splits the y’s, then (ii) and Proposi-
tion 4.15 imply that y4y5 splits the z’s, again contradicting that y4y5 splits the x’s.
Proposition 4.18. Let u be an element in {x, y, z} that satisfies the property in Propo-
sition 4.17, and suppose that u4u5 splits the v’s, where v ∈ {x, y, z} \ {u}. Then the
following hold:
(B1) u3u5 separates v5 from the other v’s;
(B2) v4v5 splits the w’s, where {w} = {x, y, z} \ {u, v};
(B3) v1v4 and v2v4 are both 7-edges; and
(B4) v3v5 separates v4 from the other v’s.
Proof. Without loss of generality, we can assume that u = x, v = y, and X,Y and Z are
placed according to Figure 11. Indeed, we can get such requirements by rotating and/or
reflecting P , and by exchanging appropriately the labels x, y and z.
(B1): From our assumptions and the hypothesis we know that x4 is the only x in x3x−5 .
Since x3x5 is an 8-edge, then exactly one element y∗ of Y is in x3x−5 . From (A6), we know
that such a y∗ is one of y4 or y5. If y∗ = y4, then, by the way y0, . . . , y5 were labelled, we
have that y5 must be contained in the triangle S of Figure 14. Then y4y5 leaves x3, x4, x5
and all the z’s on its right side. This implies that y4y5 cannot be an 8-edge, contradicting
(A4). This contradiction implies that (B1) holds.
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 373
(B2): From (B1), we know that y∗ = y5 is the only element of Y in x3x−5 . If y5 ∈
y0y
−
4 , then y4 must be contained in the region R of Figure 14. This implies that each
element in (X ∪ Y ) \ {x4, y4, y5} lies in y4y−5 , contradicting (A4) that y4y∗ = y4y5 is
an 8-edge. Thus we have that y5 ∈ y0y+4 . This fact and Proposition 4.13 imply that y4y5
splits the z’s, as required.
In view of (B2) and our previous assumptions, for the rest of the proof, we may assume
that the two triangles defined by {x0, x4, x5} and {y0, y4, y5} are as shown in Figure 12(a).
(B3): Let ℓ be the line through y4 which is parallel to x4x5 and let M be the interior of
the triangle formed by y0y5, y0y4 and x4x5. See Figure 12(a). Since y4 and y5 are the only
y’s in x4x−5 , then M ∩ Y = {y1, y2, y3}. If we rotate ℓ in clockwise order around y4 until
it reaches y4y0, then by Observation 4.7 we have that exactly one of {y1y4, y2y4, y3y4} is
8-edge and the other two are 7-edges. The desired assertion follows from (A4).
(B4): Seeking a contradiction, suppose that y3y5 does not separate y4 from the other
y’s. Then Proposition 4.16 implies that y3y5 separates {y0, y1} from {y2, y4}. On the other
hand, since y3y4 is an 8-edge and x4x5 separates y3 from y4, then y3y−4 = X ∪ {y0, y2}.
Thus y1 must be contained in the triangle R formed by z0y3, y3y4, y0y5. See Figure 12(b).
This implies that y40 = y1. Then Proposition 4.11, Observation 4.8(2) and (B3) imply,
eyy7 (P ) ≥ 6, a contradiction.
Figure 14: Here x3x5 separates y∗ from the other y’s.
Remark 4.19. From now on, without loss of generality, we assume that the u ∈ {x, y, z}
satisfying Proposition 4.17 is x, and that x5 ∈ x0x+4 . Indeed, it is not hard to see that we
can get such requirements by rotating and/or reflecting P along x0x4, and by appropriately
exchanging the labels x, y and z. In particular, we assume that X,Y, Z, x0, x4 and x5 are
placed as in Figure 11.
Note that (B4) appears as hypothesis in Propositions 4.17 and 4.18. The following
corollary is an immediate consequence of this fact.
Corollary 4.20. Let σ(x) = y, σ(y) = z and σ(z) = x. The following hold for each
u ∈ {x, y, z}:
(C1) u3u5 separates σ(u)5 from the other σ(u)’s;
(C2) σ(u)4σ(u)5 splits the σ(σ(u))’s;
374 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
(C3) σ(u)1σ(u)4 and σ(u)2σ(u)4 are both 7-edges;
(C4) σ(u)3σ(u)5 separates σ(u)4 from the other σ(u)’s; and
(C5) u5 ∈ u0u+4 .
Proof. In view of Remark 4.19, we may assume that x4x5 splits the y’s, x3x5 separates x4
from the other x’s, and X,Y, Z, x0, x4 and x5 are placed as in Figure 11.
First, we show that (C1) – (C4) hold. Proposition 4.18 states exactly (C1) – (C4) for
u = x and v = σ(x) = y. In particular, (C2) and (C4) tell us that y4y5 splits the z’s and
that y3y5 separates y4 from the other y’s, respectively. By applying Proposition 4.18 to the
last two conclusions on y’s we have that (C1) – (C4) also hold for u = y and v = σ(y) = z.
Similarly, we can conclude that (C1) – (C4) also hold for u = z and v = σ(z) = x.
Now, we show (C5). For u = x the assertion holds by Remark 4.19. We first analyze
the case u = y. From (A6) and Remark 4.19, we know that {x0, . . . , x5} lies on the left
side of both y0y4 and y0y5. Seeking a contradiction, suppose that y5 ∈ y0y−4 . Then, from
(A6) and the last two facts, it is easy to verify that x0 ∈ y3y−5 . Similarly, from y5 ∈ y0y
−
4
and (C4), we can deduce that y2 ∈ y3y−5 . Thus x0, y2 ∈ y3y
−
5 , and so Proposition 4.15
implies that y4y5 splits the x’s, contradicting (C2). An analogous argument shows that (C5)
also holds for u = z.
From Remark 4.19 and Corollary 4.20, we have that the points of P with indices 0, 3, 4
and 5 are placed as in Figure 15.
Figure 15: The relative position of the points of P with indices 0, 3, 4 and 5.
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 375
4.4 Determination of pq+ when pq is a monochromatic edge.
Lemma 3.1 for the case in which pq is a monochromatic edge will follow from Proposi-
tions 4.21 and 4.22 below. Regarding the statements of these propositions, we recall from
Remark 4.19 and Corollary 4.20 that x4x5 splits the ys, y4y5 splits the zs, and z4z5 splits
the xs.
Proposition 4.21. Let u ∈ {x, y, z}, and let v be the element in {x, y, z} \ {u} such that
u4u5 splits the vs. Then the following hold:
(D1) u0u+5 = {v0, . . . , v5};
(D2) u0u+4 = {u1, u2, u3, u5} ∪ {v0, . . . , v5};
(D3) u4u+5 = {u0, u1, u2, u3} ∪ {v0, v1, v2, v3}; and
(D4) u3u+5 = {u0, u1, u2} ∪ {v0, v1, v2, v3, v4}.
Proof. (D1): From (A7), we know that u0u5 separates the v’s from the w’s. Moreover,
because u0u5 is an edge of the convex hull of U , then u0u5 leaves the other u’s on the
same side. This and (C5) imply that {u1, u2, u3, u4} ⊂ u0u−5 . Again, from (C5) and
Proposition 4.13 we have that {v0, v1, v2, v3} ⊂ u4u+5 , and hence {v0, v1, v2, v3} ⊂ u0u
+
5 .
This and the fact that u0u5 separates the v’s from the w’s imply that {v0, . . . , v5} ⊆ u0u+5 .
We finally note that Observation 4.8(1) implies that u0u+5 = {v0, . . . , v5}, as required.
(D2): From (C5), (D1), and the way in that the points of P were labelled we have
that {v0, . . . , v5} = u0u+5 ⊂ u0u
+
4 . On the other hand, since u0u4 is an edge of the
convex hull of U , then u0u4 leaves the other u’s on the same side. This and (C5) imply
that {u1, u2, u3, u5} ⊂ u0u+4 . Thus {u1, u2, u3, u5} ∪ {v0, . . . , v5} ⊂ u0u
+
4 . Again,
Observation 4.8(1) implies that u0u+4 = {u1, u2, u3, u5} ∪ {v0, . . . , v5}, as required.
(D3): It follows immediately from (C5) and Proposition 4.13.
(D4): Clearly, {v0, . . . , v5} ∩ u4u+5 ⊂ {v0, . . . , v5} ∩ u3u
+
5 . This fact, together
with (C1) and (D3), implies that {v0, v1, v2, v3, v4} ⊂ u3u+5 . On the other hand, from
(C4) is easy to verify that {u0, . . . , u5} ∩ u3u+5 = {u0, u1, u2}. Then {u0, u1, u2} ∪
{v0, v1, v2, v3, v4} ⊂ u3u+5 . We finally note that (A4) implies that u3u
+
5 = {u0, u1, u2} ∪
{v0, v1, v2, v3, v4}, as required.
Proposition 4.22. Let u ∈ {x, y, z}, and let v be the element in {x, y, z} \ {u} such that
u4u5 splits the vs. Then the following hold:
(E1) u0u1 is an 8-edge;
(E2) u0u2 and u0u3 are 7-edges;
(E3) u2u3 and u2u5 are 8-edges;
(E4) u2u+4 = {u5} ∪ {v0, . . . , v5};
(E5) u1u+4 = {u2, u3, u5} ∪ {v0, . . . , v5} and u3u
+
4 = {u2, u5} ∪ {v0, . . . , v5};
(E6) u0u+2 = {u5} ∪ {v0, . . . , v5};
(E7) u0u+1 = {u2, u5} ∪ {v0, . . . , v5} and u0u
+
3 = {u1, u2, u5} ∪ {v0, . . . , v5};
376 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
(E8) u1u+5 = {u0} ∪ {v0, . . . , v5};
(E9) u1u2 and u1u3 are 8-edges;
(E10) u1u+3 = {u2, u5} ∪ {v0, . . . , v5};
(E11) u1u+2 = {u0, u5} ∪ {v0, . . . , v5};
(E12) u2u+5 = {u0, u1} ∪ {v0, . . . , v5}; and
(E13) u2u+3 = {u4, u5} ∪ {v0, . . . , v5}.
Proof. (E1): From Proposition 4.9, Corollary 4.10, and (A5), we know that x0x1 is a 7- or
an 8-edge. If x0x1 is a 7-edge, then x1 ∈ {x20, x40} and by Proposition 4.11, there are at
least three 7-edges of type xx involving x1 but not x0. This, Observation 4.8(2), and (C3)
imply that exx7 (P ) ≥ 6, a contradiction. Thus x0x1 must be an 8-edge.
(E2): From Observation 4.8(2), we know that there are exactly two 7-edges of the type
x0x. Since (E1) and (A6) imply that none of x0x1, x0x4, x0x5 is a 7-edge, then both x0x2
and x0x3 are 7-edges, as desired.
(E3): By Corollary 4.10 and (A5), each of x2x3 and x2x5 is a 7- or an 8-edge. From
(C3) and (E2), we know that each of x1x4, x2x4, x0x2, x0x3 is a 7-edge. Since (A2) guar-
antees the existence of an additional 7-edge involving x1 and exx7 (P ) = 5, then x2x3 and
x2x5 are 8-edges, as required.
Observation 4.23. Since z4z5 separates {x4, x5} from the other x’s (see Figure 15), then
xix4 leaves Z on its left for any i ∈ {0, 1, 2, 3}.
(E4): From (C3), we know that x1x4 and x2x4 are 7-edges. This and Observation 4.23
imply that x2x4 leaves exactly 1 or exactly 3 points of X on its left.
Suppose first that x1 ∈ x2x+4 . Since x0 ∈ x2x
−
4 , then x2x
+
4 = {x1, x3, x5} ∪ Y .
Again, Observation 4.23 implies that when we rotate x2x4 counterclockwise around x4,
the first two points that such line finds (with the tail) are x1 and x3. Since x1x4 is a 7-edge
and x3x4 is an 8-edge, then the first point that such a rotation finds must be x3, and hence
x1 ∈ x3x+4 . This and the way in which the x′s were labelled imply that x40 = x1. But then
x0x1 is a 7-edge, contradicting (E1). Then x0, x1 ∈ x2x−4 and hence x2x4 leaves exactly
three points of X on its left, namely x0, x1 and x3. The desired equality follows from the
last conclusion, Observation 4.23, and (C3).
(E5): From (A4), we know that x3x4 is an 8-edge, and from (C3) that x1x4 is a 7-edge.
Since x1, x3 ∈ x2x−4 , by (E4), then when we rotate x2x4 clockwise around x4, the first two
points that such line finds (with the tail) are precisely x1 and x3. Since x1x4 is a 7-edge and
x3x4 is an 8-edge, then such a rotation finds first x3 and then x1. The desired conclusions
are immediate from this fact and (E4).
(E6): From (E4) and the way the x’s were labelled, we know that when we rotate x2x4
clockwise around x2, the first point that such line finds (with the tail) is one of x0 or x1.
Since x0x2 is a 7-edge, by (E2), then such a point must be x0 and the desired result follows
from (E4).
(E7): From (E6), we know that the first two points that we find when we rotate x0x2
counterclockwise around x0 are x1 and x3. Since x0x1 is an 8-edge, by (E1), then we
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 377
have that such a rotation finds x1 and then x3. These together with (E6) imply the desired
results.
(E8): (D4) implies that x3 ∈ x1x−5 . If x2 ∈ x1x
+
5 , then {x1, x3, x4} ∪ Z ⊂ x2x
−
5 .
This would imply that x2x5 is not an 8-edge, contradicting (E3). Then we can assume that
x2 ∈ x1x−5 . Thus, x0 is the only x on the right side of x1x5 and hence x1x5 is a (≤ 7)-
edge. From Proposition 4.9 and Corollary 4.10, we have that x1x5 must be a 7-edge. This
implies that Y ⊂ x1x+5 , and hence (E8) holds.
(E9): (E4), (E5), (E6), (E7), and (E8) imply, respectively, that x2x4, x1x4, x0x2, x0x3
and x1x5 are 7-edges. Then Proposition 4.9 implies that these five are all the monochro-
matic edges of type xx. This and Corollary 4.10 imply that x1x2 and x1x3 must be 8-edges.
(E10): The first assertion of (E7) implies that x3, x4 ∈ x0x−1 and x2, x5 ∈ x0x
+
1 .
From the first assertion of (E5) we know that x2, x3 ∈ x1x+5 . Then when we rotate x0x1
counterclockwise around x1, the first point that such line finds must be x3, and so the
desired result follows immediately from this and the first assertion of (E5).
(E11): From the first assertion of (E7), we have that x3, x4 ∈ x0x−1 and x2, x5 ∈ x0x
+
1 .
From (E8), we know that x2, x3 ∈ x1x−4 . Then when we rotate x0x1 clockwise around x1,
the first point that we find is x2, and so the desired result follows from the first assertion of
(E7).
(E12): (E8) implies {x2, x3, x4} ∪ Z = x1x−5 . From (D4) and the second assertion of
(E3), we know that when we rotate x1x5 clockwise around x5, the first point that we find
must be x2, and so the desired result follows from (E8).
(E13): (E4) implies that Z ∪ {x0, x1, x3} = x2x−4 . From (E10), we know that x2 ∈
x1x
+
3 . This and the first assertion of (E3) imply that when we rotate x2x4 counterclockwise
around x2, the first point that we find must be x3, and so the desired result follows from
(E4).
4.5 Determination of pq+ when pq is a bichromatic (>5)-edge.
We are finally ready to prove Lemma 3.1 for the remaining case, namely when pq is a
bichromatic (> 5)-edge. This is achieved in the next statement. We recall from Re-
mark 4.19 and Corollary 4.20 that x4x5 splits the ys, y4y5 splits the zs, and z4z5 splits
the xs.
Proposition 4.24. Let u ∈ {x, y, z}, and let v be the element in {x, y, z} \ {u} such that
u4u5 splits the vs. Then the following hold:
(F1) u5v+4 = {u0, u1, u2, u3} ∪ {v0, v1, v2, v3};
(F2) u5v+5 = {u0, u1, u2} ∪ {v0, v1, v2, v3, v4};
(F3) u5v+3 = {u0, u1, u2, u3, u4} ∪ {v0, v1, v2};
(F4) u5v+2 = {u0, u1, u2, u3, u4} ∪ {v0, v1};
(F5) u5v+1 = {u0, u1, u2, u3, u4} ∪ {v0};
(F6) u4v+4 = {u0, u1, u2, u3, u5} ∪ {v0, v1, v2, v3};
(F7) u4v+5 = {u0, u1, u2, u3, u5} ∪ {v0, v1, v2, v3, v4};
378 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
(F8) u4v+3 = {u0, u1, u2, u3} ∪ {v0, v1, v2};
(F9) u4v+2 = {u0, u1, u2, u3} ∪ {v0, v1};
(F10) u3v+5 = {u0, u1, u2, u5} ∪ {v0, v1, v2, v3, v4};
(F11) u3v+4 = {u0, u1, u2} ∪ {v0, v1, v2, v3};
(F12) u3v+3 = {u0, u1, u2} ∪ {v0, v1, v2};
(F13) u2v+5 = {u0, u1} ∪ {v0, v1, v2, v3, v4};
(F14) u2v+4 = {u0, u1} ∪ {v0, v1, v2, v3};
(F15) u1v+5 = {u0} ∪ {v0, v1, v2, v3, v4}.
Proof. In view of Remark 4.19 and Corollary 4.20, we may assume that the points of P are
placed as in Figure 15. Moreover, by symmetry, we only need to verify the case u = x and
v = y. For brevity, form ∈ {0, . . . , 5}, we letXm := {xi
∣∣i ≤ m} and Ym := {yi∣∣i ≤ m}.
(F1): From (D3), we know that x4x+5 = X3 ∪ Y3. We claim that the first point p ∈ P
that x4x5 finds when it is rotated counterclockwise around x5 is y4. From (D4), we know
that x3x+5 = X2 ∪ Y4. Then p = y4, and so x5y
+
4 = X3 ∪ Y3, as required.
(F2): We know that x3x+5 = X2 ∪ Y4 by (D4). We claim that the first point p ∈ P that
x3x5 finds when it is rotated counterclockwise around x5 is y5. Indeed, from (D1), (E8),
and (E12), we know that y5 ∈ xjx+5 for j = 0, 1, 2, respectively. These imply that p /∈ X2,
and so p = y5. Then x5y+5 = X2 ∪ Y4, as required.
(F3): We know that x4x+5 = X3 ∪ Y3 by (D3). We claim that the first point p ∈ P
that x4x5 finds when it is rotated clockwise around x5 is y3. Indeed, by applying (E7),
(E10), and (E13) to j = 0, 1 and j = 2 (with u = y and v = z), respectively, we have that
X ⊂ yjy−3 , and so x5 ∈ yjy
−
3 . These imply that p /∈ Y2. This and the fact that x5y
+
0 = X4
imply that p = y3. Then x5y+3 = X4 ∪ Y2, as required.
(F4): We know that x5y+3 = X4 ∪ Y2 by (F3). We claim that the first point p ∈ P
that x5y3 finds when it is rotated clockwise around x5 is y2. Indeed, by applying (E6) and
(E11) to j = 0 and j = 1 (with u = y and v = z), respectively, we have that X ⊂ yjy−2 ,
and so x5 ∈ yjy−2 . These imply that p /∈ Y1. This and the fact that x5y
+
0 = X4 imply that
p = y2. Then x5y+2 = X4 ∪ Y1, as required.
(F5): We know that x5y+2 = X4 ∪ Y1 by (F4). We claim that the first point p ∈ P that
x5y2 finds when it is rotated clockwise around x5 is y1. Indeed, by taking u = y in (E7),
we have that x5 ∈ y0y−1 , and so p ̸= y0. This and the fact that x5y
+
0 = X4 imply that
p = y1. Then x5y+1 = X4 ∪ Y0, as required.
(F6): We know that x4x+5 = X3 ∪ Y3 by (D3). We claim that the first point p ∈ P
that x4x5 finds when it is rotated counterclockwise around x4 is y4. Indeed, by taking
u = y and v = z in D3) we get x4 ∈ y4y−5 , and so p ̸= y5. This and the fact that
x0x
+
4 = {x1, x2, x3, x5} ∪ Y imply that p = y4. Then x4y
+
4 = X3 ∪ Y3 ∪ {x5}, as
required.
(F7): We know that x4y+4 = X3 ∪ Y3 ∪ {x5} by (F6). We claim that the first point
p ∈ P that x4y4 finds when it is rotated counterclockwise around x4 is y5. Indeed, from
(D2), we know that x0x+4 = {x1, x2, x3, x5}∪Y . These imply that p /∈ X , and so p = y5.
Then x4y+5 = X3 ∪ Y4 ∪ {x5}, as required.
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 379
(F8): We know that x4x+5 = X3 ∪ Y3 by (D3). We claim that the first point p ∈ P that
x4x5 finds when it is rotated clockwise around x4 is y3. Indeed, by applying E7), E10), and
E13) to j = 0, 1 and j = 2 (with u = y and v = z), respectively, we have that X ⊂ yjy−3 ,
and so x4 ∈ yjy−3 . These imply that p /∈ Y2. From Proposition 4.3(2), we know that
X3 ⊂ x4y+1 , and so p /∈ X3. All these facts imply that p = y3, and so x4y
+
3 = X3 ∪Y2, as
required.
(F9): We know that x4y+3 = X3 ∪ Y2 by (F8). We claim that the first point p ∈ P
that x4y3 finds when it is rotated clockwise around x4 is y2. Indeed, by applying (E6) and
(E11) to j = 0 and j = 1 (with u = y and v = z), respectively, we have that X ⊂ yjy−2 ,
and so x4 ∈ yjy−2 . These imply that p /∈ Y1. From this and (D3) it follows that p = y2,
and so x4y+2 = X3 ∪ Y1, as required.
(F10): We know that x3x+5 = X2∪Y4 by (D4). We claim that the first point p ∈ P that
x3x5 finds when it is rotated counterclockwise around x3 is y5. Indeed, by applying (E7),
(E10), and (E13) to j = 0, 1 and j = 2 (with u = x and v = y), respectively, we have that
Y ⊂ xjx+3 , and so y5 ∈ xjx
+
3 . These imply that p /∈ X2. From this and E5) it follows that
p = y5, and so x3y+5 = X2 ∪ Y4 ∪ {x5}, as required.
(F11): We know that x3x+5 = X2 ∪ Y4 by D4). We claim that the first point p ∈ P that
x3x5 finds when it is rotated clockwise around x3 is y4. Indeed, by applying (D2), (E5),
(E4), and (E5) to j = 0, 1, 2 and j = 3 (with u = y and v = z), respectively, we have
that X ⊂ yjy−4 , and so x3 ∈ yjy
−
4 . These imply that p /∈ Y3. From Proposition 4.3(2),
we know that x4 ∈ x3y−2 , and so p ̸= x4. All these facts imply that p = y4, and so
x3y
+
4 = X2 ∪ Y3, as required.
(F12): We know that x3y+4 = X2 ∪ Y3 by (F11). We claim that the first point p ∈ P
that x3y4 finds when it is rotated clockwise around x3 is y3. Indeed, by applying (E7),
(E10), and (E13) to j = 0, 1 and j = 2 (with u = y and v = z), respectively, we have that
X ⊂ yjy−3 , and so x3 ∈ yjy
−
3 . These imply that p /∈ Y2. By taking u = x in (E5) and (D4),
we have that y4 ∈ x3x+4 and y4 ∈ x3x
+
5 , respectively. These imply that p /∈ {x4, x5}. All
these facts imply that p = y3, and so x3y+3 = X2 ∪ Y2, as required.
(F13): From Proposition 4.3(2), we know that x2y+3 = X1 ∪Y2. We claim that the first
point p ∈ P that x2y3 finds when it is rotated counterclockwise around x2 is y4. Indeed,
by applying (E6) and (E11) to j = 0 and j = 1 (with u = x and v = y), respectively, we
have that Y ⊂ xjx+2 , and so p /∈ {x0, x1}. Finally, by applying (E4), (E12), and (E13) to
j = 4, 5 and j = 3 (with u = x and v = y), we have that p ̸= x4, p ̸= x5 and p ̸= x3,
respectively. All these facts imply that p = y4, and so x2y+4 = X1 ∪ Y3, as required.
(F14): We know that x2y+4 = X1 ∪ Y3 by (F13). We claim that the first point p ∈ P
that x2y4 finds when it is rotated counterclockwise around x2 is y5. As in (F13) we can
deduce from (E6) and (E11) that p /∈ {x0, x1}. Again, as in (F13) we can deduce from
(E4), (E12), and (E13) that p ̸= x4, p ̸= x5 and p ̸= x3, respectively. All these facts imply
that p = y5, and so x2y+5 = X1 ∪ Y4, as required.
(F15): We know that x2y+5 = X1 ∪ Y4 by F14). We claim that the first point p ∈ P
that x2y5 finds when it is rotated counterclockwise around y5 is x1. Indeed, from Proposi-
tion 4.3(2), we know that y5z+0 = Y4. From the last two equations we have that p ∈ X1 =
{x0, x1}. Again, from Proposition 4.3(2), we know that x0y+5 = Y4, and so p = x1. Then
x1y
+
5 = X0 ∪ Y4, as required.
380 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
4.6 Conclusion of the proof of Lemma 3.1
In Tables 1 and 2 we give a summary of the results in Propositions 4.3(2)(b), 4.21, 4.22,
and 4.24. These tables assume that u ∈ {x, y, z} and v = σ(u), where σ is the automor-
phism of {x, y, z} defined in Corollary 4.20, namely x σ7→ y, y σ7→ z, and z σ7→ x.
In particular, for each u ∈ {x, y, z} and m,n ∈ {0, . . . , 5} with m < n, the set umu+n
is given in Table 1. This also determines the set unu+m, since unu
+
m = umu
−
n , and umu
−
n
is evidently determined from umu+n . Thus the information in Table 1 suffices to determine
pq+ whenever pq is a monochromatic edge of P .
Now for each u ∈ {x, y, z} and each m,n ∈ {0, . . . , 5}, the set umv+n is given in
Table 2. This also determines the set vnu+m, since vnu
+
m = umv
−
n , and umv
−
n is evidently
determined from umv+n . Thus the information in Table 2 suffices to determine pq
+ when-
ever pq is a bichromatic edge of P . □
uiu
+
j for each uiuj ∈ Emonk (P ) Classification of uiuj Equality stated in
u0u
+
5 = {v0, . . . , v5} 6-edge (D1)
u0u
+
4 = {u1, u2, u3, u5} ∪ {v0, . . . , v5} 6-edge (D2)
u0u
+
3 = {u1, u2, u5} ∪ {v0, . . . , v5} 7-edge (E7)
u0u
+
2 = {u5} ∪ {v0, . . . , v5} 7-edge (E6)
u0u
+
1 = {u2, u5} ∪ {v0, . . . , v5} 8-edge (E7)
u1u
+
5 = {u0} ∪ {v0, . . . , v5} 7-edge (E8)
u1u
+
4 = {u2, u3, u5} ∪ {v0, . . . , v5} 7-edge (E5)
u1u
+
3 = {u2, u5} ∪ {v0, . . . , v5} 8-edge (E10)
u1u
+
2 = {u0, u5} ∪ {v0, . . . , v5} 8-edge (E11)
u2u
+
5 = {u0, u1} ∪ {v0, . . . , v5} 8-edge (E12)
u2u
+
4 = {u5} ∪ {v0, . . . , v5} 7-edge (E4)
u2u
+
3 = {u4, u5} ∪ {v0, . . . , v5} 8-edge (E13)
u3u
+
5 = {u0, u1, u2} ∪ {v0, . . . , v4} 8-edge (D4)
u3u
+
4 = {u2, u5} ∪ {v0, . . . , v5} 8-edge (E5)
u4u
+
5 = {u0, . . . , u3} ∪ {v0, . . . , v3} 8-edge (D3)
Table 1: All the monochromatic edges of P .
5 Concluding remarks
In this work we finally have given the full proof of Theorem 1.2, which was announced at
the EuroComb’11 conference [1].
As we mentioned in the Introduction, the exact rectilinear crossing number of Kn is
known only for n ≤ 27 and n = 30 [3, 7, 8, 9, 10]. In [2] and [6] we can find non-
isomorphic crossing-minimal rectilinear drawings of Kn for both n = 24 and n = 30.
On the other hand, from [6] and the main result of this work, now we know that there
is a unique (up to order type isomorphism) crossing-minimal rectilinear drawing of Kn,
for n = 6, 12, 18. Thus, a plausible conjecture is that K6m has several crossing-minimal
rectilinear drawings for each integer m ≥ 4.
We close this paper with a discussion on a question raised by an anonymous reviewer
of an earlier version of this paper: to what degree would it be possible to get a computer-
assisted proof of Theorem 1.2?
B. M. Ábrego et al.: There is a unique crossing-minimal rectilinear drawing of K18 381
uv+ for each uv ∈ Ebik (P ) Classification of uv Equality stated in
u0v
+
5 = {v0, . . . , v4} 5-edge Proposition 4.3(2)
u0v
+
4 = {v0, v1, v2, v3} 4-edge Proposition 4.3(2)
u0v
+
3 = {v0, v1, v2} 3-edge Proposition 4.3(2)
u0v
+
2 = {v0, v1} 2-edge Proposition 4.3(2)
u0v
+
1 = {v0} 1-edge Proposition 4.3(2)
u0v
+
0 = ∅ 0-edge Proposition 4.3(2)
u1v
+
5 = {u0} ∪ {v0, . . . , v4} 6-edge (F15)
u1v
+
4 = {u0} ∪ {v0, v1, v2, v3} 5-edge Proposition 4.3(2)
u1v
+
3 = {u0} ∪ {v0, v1, v2} 4-edge Proposition 4.3(2)
u1v
+
2 = {u0} ∪ {v0, v1} 3-edge Proposition 4.3(2)
u1v
+
1 = {u0} ∪ {v0} 2-edge Proposition 4.3(2)
u1v
+
0 = {u0} 1-edge Proposition 4.3(2)
u2v
+
5 = {u0, u1} ∪ {v0, . . . , v4} 7-edge (F14)
u2v
+
4 = {u0, u1} ∪ {v0, v1, v2, v3} 6-edge (F13)
u2v
+
3 = {u0, u1} ∪ {v0, v1, v2} 5-edge Proposition 4.3(2)
u2v
+
2 = {u0, u1} ∪ {v0, v1} 4-edge Proposition 4.3(2)
u2v
+
1 = {u0, u1} ∪ {v0} 3-edge Proposition 4.3(2)
u2v
+
0 = {u0, u1} 2-edge Proposition 4.3(2)
u3v
+
5 = {u0, u1, u2, u5} ∪ {v0, . . . , v4} 7-edge (F10)
u3v
+
4 = {u0, u1, u2} ∪ {v0, v1, v2, v3} 7-edge (F11)
u3v
+
3 = {u0, u1, u2} ∪ {v0, v1, v2} 6-edge (F12)
u3v
+
2 = {u0, u1, u2} ∪ {v0, v1} 5-edge Proposition 4.3(2)
u3v
+
1 = {u0, u1, u2} ∪ {v0} 4-edge Proposition 4.3(2)
u3v
+
0 = {u0, u1, u2} 3-edge Proposition 4.3(2)
u4v
+
5 = {u0, u1, u2, u3, u5} ∪ {v0, . . . , v4} 6-edge (F7)
u4v
+
4 = {u0, u1, u2, u3, u5} ∪ {v0, . . . , v3} 7-edge (F6)
u4v
+
3 = {u0, u1, u2, u3} ∪ {v0, v1, v2} 7-edge (F8)
u4v
+
2 = {u0, u1, u2, u3} ∪ {v0, v1} 6-edge (F9)
u4v
+
1 = {u0, u1, u2, u3} ∪ {v0} 5-edge Proposition 4.3(2)
u4v
+
0 = {u0, u1, u2, u3} 4-edge Proposition 4.3(2)
u5v
+
5 = {u0, u1, u2} ∪ {v0, . . . , v4} 8-edge (F2)
u5v
+
4 = {u0, . . . , u3} ∪ {v0, . . . , v3} 8-edge (F1)
u5v
+
3 = {u0, . . . , u4} ∪ {v0, v1, v2} 8-edge (F3)
u5v
+
2 = {u0, . . . , u4} ∪ {v0, v1} 7-edge (F4)
u5v
+
1 = {u0, . . . , u4} ∪ {v0} 6-edge (F5)
u5v
+
0 = {u0, u1, u2, u3, u4} 5-edge Proposition 4.3(2)
Table 2: All the bichromatic edges of P .
382 Ars Math. Contemp. 24 (2024) #P2.10 / 355–383
At the beginning of this project we asked ourselves the same question, but we are con-
vinced that a traditional proof might be easier to verify. It is worth mentioning that a heav-
ily computer-assisted proof seems to be out of reach, most likely involving several hundred
million CPU hours. On the other hand, we believe that a partially computer-assisted proof
would be more difficult to follow and perhaps also less reliable. Using computer-assisted
proofs needs a very careful preparation and description of what is done, and proofs of the
correctness of the results. The code must be explained in full detail, as well as how the
program can be executed (including the operating system, compiler versions, etc.). We
believe that in this particular case the task of verifying all this information would end up
being more taxing on the reader than the current purely theoretical proof.
ORCID iDs
Bernardo M. Ábrego https://orcid.org/0000-0003-4695-5454
Silvia Fernández–Merchant https://orcid.org/0000-0003-2080-106X
Oswin Aichholzer https://orcid.org/0000-0002-2364-0583
Jesús Leaños https://orcid.org/0000-0002-3441-8136
Gelasio Salazar https://orcid.org/0000-0002-8458-3930
References
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10.1016/j.endm.2011.09.089, https://doi.org/10.1016/j.endm.2011.09.089.
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