Proper Ehresmann semigroups

TL;DR

This paper introduces proper Ehresmann semigroups, providing a structural framework based on graph-labelled generators, and demonstrates their relation to existing algebraic structures and covers.

Contribution

It defines proper Ehresmann semigroups via a three-coordinate description and establishes their structure, including the existence of proper covers and connections to partial multiactions.

Findings

Every Ehresmann semigroup has a proper cover.

Proper Ehresmann semigroups with three-coordinate elements are characterized by partial multiactions.

The covering monoid aligns with previous work by Branco, Gomes, and Gould.

Abstract

We propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and $\sigma$-class, where $\sigma$ denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann semigroups and show that every Ehresmann semigroup has a proper cover. Our covering monoid turns out to be isomorphic to that from the work by Branco, Gomes and Gould and provides a new view of the latter. Proper Ehresmann semigroups all of whose elements admit a three-coordinate description are characterized in terms of partial multiactions of monoids on semilattices. As a consequence we recover the two-coordinate structure result on proper restriction semigroups.