Coherency properties for monoids of transformations and partitions

Abstract

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act itself has a finite presentation; it is weakly right coherent if every finitely generated right ideal of $S$ has a finite presentation. We show that full and partial transformation monoids, symmetric inverse monoids and partition monoids over an infinite set are all weakly right coherent, but that none of them is right coherent. Left coherency and weak left coherency are defined dually, and the corresponding results hold for these properties. In order to prove the non-coherency results, we give a presentation of an inverse semigroup which does not embed into any left or right coherent monoid.