ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.02
https://doi.org/10.26493/2590-9770.1259.9d5
(Also available at http://adam-journal.eu)
A graph-theoretic method to define any Boolean
operation on partitions
David P. Ellerman ∗
Philosophy Department, University of California, Riverside, CA 92521 USA
and School of Social Sciences, University of Ljubljana, Slovenia
Received 4 August 2018, accepted 23 August 2018, published online 10 June 2019
Abstract
The lattice operations of join and meet were defined for set partitions in the nineteenth
century, but no new logical operations on partitions were defined and studied during the
twentieth century. Yet there is a simple and natural graph-theoretic method presented
here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic
method is also defined. In closing, the question is addressed of why it took so long for all
Boolean operations to be defined for partitions.
Keywords: Set partitions, Boolean operations, graph-theoretic methods, closure-theoretic methods.
Math. Subj. Class.: 05A18, 03G10
∗ORCID iD: 0000-0002-5718-618X
E-mail address: david@ellerman.org (David P. Ellerman)
cb This work is licensed under https://creativecommons.org/licenses/by/3.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.02
https://doi.org/10.26493/2590-9770.1259.9d5
(Dostopno tudi na http://adam-journal.eu)
Metoda teorije grafov za definiranje poljubne
Booleove operacije na particijah
David P. Ellerman ∗
Philosophy Department, University of California, Riverside, CA 92521 USA
and School of Social Sciences, University of Ljubljana, Slovenia
Prejeto 4. avgusta 2018, sprejeto 23. avgusta 2018, objavljeno na spletu 10. junija 2019
Povzetek
Mrežni operaciji unija in presek sta bili definirani za particije množic v devetnajstem
stoletju, a v dvajsetem stoletju niso bile definirane in raziskovane nobene nove logične o-
peracije na particijah. Vendar obstaja preprosta in naravna metoda teorije grafov za defini-
ranje poljubne Booleove operacije na n-tericah particij in ta metoda je predstavljena v
tem članku. Definiramo tudi ekvivalentno metodo teorije zaprtij. Nazadnje obravnavamo
vprašanje, zakaj je trajalo tako dolgo, da so bile vse Booleove operacije definirane na par-
ticijah.
Ključne besede: Particije množice, Booleove operacije, metode teorije grafov, metode teorije zaprtij.
Math. Subj. Class.: 05A18, 03G10
∗ORCID iD: 0000-0002-5718-618X
E-poštni naslov: david@ellerman.org (David P. Ellerman)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/3.0/