Left Ehresmann monoids with a proper basis

TL;DR

This paper develops a theory for left Ehresmann monoids, introducing proper bases, and shows their structural properties and classifications, drawing analogies to inverse semigroup theory.

Contribution

It introduces the notion of proper bases for left Ehresmann monoids and characterizes those with proper bases as isomorphic to specific subsemigroups, extending inverse semigroup results.

Findings

Left Ehresmann monoids with proper bases have properties similar to two-sided Ehresmann monoids.

Any such monoid is isomorphic to a subsemigroup  of a constructed class .

A globalization result for order-preserving partial monoid actions is presented.

Abstract

Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for left Ehresmann monoids inspired by that for inverse semigroups; in order to do so we must develop substantially different ideas and techniques. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$, and consequently any free left Ehresmann monoid, possesses a proper basis. We show that any left Ehresmann monoid with a proper basis displays properties close to those of two-sided Ehresmann monoids. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$, which are also left Ehresmann with a proper basis. We prove that any left Ehresmann monoid with a proper basis is isomorphic to some $\mathcal{Q}_{\ell}(T,X,Y)$. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group. In the process of proving our main theorems we present a globalisation result for an order-preserving partial action of a monoid on a partially ordered set or semilattice.