ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.03
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/
2 Ars Math. Contemp. 24 (2024) #P2.03
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 3
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),
4 Ars Math. Contemp. 24 (2024) #P2.03
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 5
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).
6 Ars Math. Contemp. 24 (2024) #P2.03
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 7
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 .
8 Ars Math. Contemp. 24 (2024) #P2.03
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 9
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.
10 Ars Math. Contemp. 24 (2024) #P2.03
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 11
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.
12 Ars Math. Contemp. 24 (2024) #P2.03
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 13
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.
14 Ars Math. Contemp. 24 (2024) #P2.03
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 15
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.
16 Ars Math. Contemp. 24 (2024) #P2.03
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 17
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).
18 Ars Math. Contemp. 24 (2024) #P2.03
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 19
(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.
20 Ars Math. Contemp. 24 (2024) #P2.03
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 21
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.
22 Ars Math. Contemp. 24 (2024) #P2.03
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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