On homological finiteness properties and free inverse monoids

Abstract

We construct a simple and useful sufficient condition, based on actions on a lattice of idempotents, for monoids admitting homomorphisms to the monogenic free inverse monoid FIM(1) to not be of type FP2. This recovers a result of Gray and Steinberg that free inverse monoids are not of type FP2. The same technique is then used to show that a finitely generated submonoid of FIM(1) is of type FP2 if and only if it is finitely presented, answering a question of Cho & Ruškuc.

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