Subset expansions of monoids

TL;DR

This paper introduces the subset expansion of monoids, exploring its algebraic properties and finitary conditions, revealing how it transforms various classes of monoids and characterizing when certain properties are preserved or lifted.

Contribution

It defines the subset expansion of monoids and investigates its algebraic and finitary properties, providing characterizations and conditions for property preservation and lifting.

Findings

Maps groups to proper inverse monoids

Preserves certain finitary conditions under retracts

Characterizes when properties like property (L) are satisfied

Abstract

We initiate the study of the expansion $\mathcal{S}(M)$ of a monoid $M$ obtained via the semidirect product of $M$ acting naturally on the left of its power set (regarded as a semilattice under union). We term this the `subset expansion' of $M$. The monoid $\mathcal{S}(M)$ contains the images of several expansions of $M$ of wide interest and use in semigroup theory, in particular the prefix and Szendrei expansions (in the case where $M$ is free, these `smaller' expansions produce free algebras in certain varieties). We first focus on algebraic properties, specifically those determined by idempotents. Particularly, we show that the expansion $\mathcal{S}$ maps groups to proper inverse monoids, unipotent monoids to proper left restriction monoids, right cancellative monoids to left ample monoids, right abundant monoids to right abundant monoids, and left cancellative monoids to right adequate monoids. Subsequently, we focus on finitary conditions. We examine the condition of weak left coherence (every finitely generated left ideal has a finite presentation as a left act); the related conditions of property (L), left ideal Howson, finitely left equated, and each of the corresponding left-right dual notions. Each of these conditions is preserved under retract, from which it is immediate that if $\mathcal{S}(M)$ satisfies one of our finitary conditions, then so must $M$, but the converse is not true. For a property to `lift' from $M$ to $\mathcal{S}(M)$ it must undergo a strengthening. Indeed, we show that $\mathcal{S}(M)$ satisfies property (L) (or its left-right dual) if and only if $M$ is finite. We provide exact characterisations of the monoids $M$ such that $\mathcal{S}(M)$ is: left (or right) ideal Howson; finitely left equated; and (consequently) weakly left coherent. We give sufficient conditions for $\mathcal{S}(M)$ to be finitely right equated and hence weakly right coherent.