Ehresmann theory and partition monoids

James East, Robert Gray

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This article concerns Ehresmann structures in the partition monoid P X. Since P X contains the symmetric and dual symmetric inverse monoids on the same base set X, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on P X while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): “surjections between subsets” for relations become “injections between quotients” for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems.

Original languageEnglish
Pages (from-to)318-352
Number of pages35
JournalJournal of Algebra
Volume579
Early online date1 Apr 2021
DOIs
Publication statusPublished - 1 Aug 2021

Keywords

  • Dual symmetric inverse monoids
  • Ehresmann categories
  • Ehresmann monoids
  • Partition monoids
  • Relation monoids
  • Symmetric inverse monoids

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