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A monoid $S$ is right coherent if every finitely generated subact of everyfinitely presented right $S$-act is finitely presented. The correspondingnotion for a ring $R$ states that every finitely generated submodule of everyfinitely presented right $R$-module is finitely presented. For monoids (andrings) right coherency is a finitary property which determines the existence ofa model companion of the class of right $S$-acts (right $R$-modules) and hencethat the class of existentially closed right $S$-acts (right $R$-modules) isaxiomatisable.  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 objectiveof 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 freeinverse and free left ample monoids satisfy $({\bf R})$, $({\bf r)}$, $({\bfL})$ and $({\bf l)}$, conditions arising from the axiomatisability of classesof right $S$-acts and of left $S$-acts.

Coherency, free inverse monoids and free left ample monoids