Coherency for monoids and purity for their acts

TL;DR

This paper explores the relationship between right coherency of monoids, solutions of equations over their acts, and injectivity properties, establishing key equivalences and conditions for purity concepts in monoid acts.

Contribution

It proves that for right coherent monoids, almost pure and absolutely pure acts coincide, and characterizes right coherency via classes of pure acts.

Findings

In right coherent monoids, almost pure and absolutely pure acts are the same.

A monoid is right coherent iff mfp-pure and absolutely pure acts coincide.

Examples of non-coherent monoids where pure classes coincide are provided.

Abstract

This article examines the three-way relationship between right coherency of a monoid $S$, solutions of equations over $S$-acts, and injectivity properties of $S$-acts. A monoid $S$ is right coherent if every finitely generated subact of every finitely presented (right) $S$-act itself has a finite presentation. Purity properties of an $S$-act $A$ may either be expressed in terms of solutions in $A$ of certain consistent sets of equations over $A$, or in terms of injectivity properties. For example, an $S$-act $A$ is absolutely pure (almost pure) if every finite consistent set of equations over $A$ (in one variable) has a solution in $A$. Equivalently, $A$ is absolutely pure (almost pure) if it is injective with respect to inclusions of finitely generated subacts into finitely presented (monogenic finitely presented) $S$-acts. Our first main result shows that for a right coherent monoid $S$ the classes of almost pure and absolutely pure $S$-acts coincide. Our second main result is that a monoid $S$ is right coherent if and only if the classes of mfp-pure and absolutely pure $S$-acts coincide: an $S$-act is mfp-pure if it is injective with respect to inclusions of finitely presented subacts into monogenic finitely presented $S$-acts. We give specific examples of monoids $S$ that are not right coherent yet are such that the classes of almost pure and absolutely pure $S$-acts coincide. Finally we give a condition on a monoid $S$ for all almost pure $S$-acts to be absolutely pure in terms of finitely presented $S$-acts, their finitely generated subacts, and certain canonical extensions.