Homological finiteness properties of monoids, their ideals and maximal subgroups

TL;DR

This paper investigates how homological finiteness properties in monoids relate to their substructures, establishing inheritance and equivalence conditions for properties like left-FPn in ideals and maximal subgroups.

Contribution

It provides new results on the inheritance and equivalence of homological finiteness properties between monoids, their ideals, and maximal subgroups, especially in the context of completely simple semigroups.

Findings

Left-FPn is inherited by maximal subgroups in certain minimal ideals.

A completely simple semigroup is of type left- and right-FPn iff it has finitely many ideals and all maximal subgroups are of type FPn.

If an ideal has a two-sided identity, the monoid is of type left-FPn iff the ideal is of that type.

Abstract

We consider the general question of how the homological finiteness property left-FPn holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. In particular we show that left-FPn is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups we prove the converse, and as a corollary show that a completely simple semigroup is of type left- and right-FPn if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type FPn. Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left-FPn if and only if the ideal is of type left-FPn.