Ehresmann theory and partition monoids

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.

Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

“…Guo and Chen [7] obtained a similar result for finite ample semigroups and the author extended this generalization to a class of right restriction E-Ehresmann semigroups [19,20] (E-Ehresmann semigroups were introduced by Lawson in [11]). This result has led to several applications regarding semigroups of partial functions [18,21,22,13] and recently also to the study of certain partition monoids [3]. We mention also that Wang [26] generalized the above results further to a certain class of right P -restriction, P -Ehresmann semigroups (for definitions of these notions see [10]) -but we do not follow this approach in this paper.…”

On Ehresmann semigroups

“…Ehresmann semigroups have emerged as an interesting class [1,2,4,15]. In particular, they are closely allied to categories in two ways.…”