Coherency for monoids and purity for their acts

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.