Ehresmann theory and partition monoids

TL;DR

This paper investigates Ehresmann structures within the partition monoid, revealing how certain semilattices induce these structures and exploring their implications for submonoids, dualities, and related diagram monoids.

Contribution

It identifies which semilattices in the partition monoid induce Ehresmann structures and characterizes maximal restriction submonoids, contrasting these findings with relation monoids.

Findings

One semilattice induces an Ehresmann structure on $P_X$.

Characterization of largest restriction submonoids.

Dualities between relation 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.