Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version

TL;DR

This paper investigates the representation theory of a class of Ehresmann semigroups with EI-categories, providing descriptions of simple and projective modules, and analyzing their structure through examples like the monoid of partial functions.

Contribution

It characterizes simple and projective modules of EI-category Ehresmann semigroups, extending understanding of their algebraic structure and representation theory.

Findings

Simple modules induced from maximal subgroups via Schützenberger modules.

Indecomposable projective modules described using generalized Green's relations.

Explicit formulas for projective modules and Cartan matrix entries for the monoid of partial functions.

Abstract

We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formed by inducing the simple modules of the maximal subgroups of $S$ via the corresponding Sch\"{u}tzenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid $\mathcal{PT}_{n}$ of all partial functions on an $n$-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.