Coherency, free inverse monoids and free left ample monoids

Abstract

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. The corresponding notion for a ring $R$ states that every finitely generated submodule of every finitely presented right $R$-module is finitely presented. For monoids (and rings) right coherency is a finitary property which determines the existence of a model companion of the class of right $S$-acts (right $R$-modules) and hence that the class of existentially closed right $S$-acts (right $R$-modules) is axiomatisable. Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruskuc, have shown that groups, and free monoids, have the same properties. We demonstrate that free inverse monoids do not. Any free inverse monoid contains as a submonoid the free left ample monoid, and indeed the free monoid, on the same set of generators. The main objective of the paper is to show that the free left ample monoid is right coherent. Furthermore, by making use of the same techniques we show that both free inverse and free left ample monoids satisfy $({\bf R})$, $({\bf r)}$, $({\bf L})$ and $({\bf l)}$, conditions arising from the axiomatisability of classes of right $S$-acts and of left $S$-acts.