Homotopy bases and finite derivation type for subgroups of monoids

TL;DR

This paper develops methods to derive homotopy bases for subgroups of monoids, showing conditions under which subgroups inherit finite derivation type, with applications to regular monoids and their maximal subgroups.

Contribution

It introduces a way to construct homotopy bases for subgroups of monoids and establishes conditions for these subgroups to have finite derivation type, extending known results.

Findings

Subgroups inherit FDT under certain finiteness conditions.

Regular monoids with finitely many ideals have FDT if all maximal subgroups do.

Finitely presented regular monoids satisfy FP_3 if their maximal subgroups do.

Abstract

Given a monoid defined by a presentation, and a homotopy base for the derivation graph associated to the presentation, and given an arbitrary subgroup of the monoid, we give a homotopy base (and presentation) for the subgroup. If the monoid has finite derivation type (FDT), and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite, then we obtain a finite homotopy base for the subgroup, and hence the subgroup has FDT. As an application we prove that a regular monoid with finitely many left and right ideals has FDT if and only if all of its maximal subgroups have FDT. We use this to show that a finitely presented regular monoid with finitely many left and right ideals satisfies the homological finiteness condition FP_3 if all of its maximal subgroups satisfy the condition FP_3.