Homological finiteness properties of monoids, their ideals and maximal subgroups

We consider the general question of how the homological finiteness property left-(resp. right-) holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. This is done by giving methods for constructing free resolutions of substructures from free resolutions of their containing monoids, and vice versa. In particular, we show that left- 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- if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type . 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- if and only if the ideal is of type left-. Applying this result, we obtain necessary and sufficient conditions for a Clifford monoid (and more generally a strong semilattice of monoids) to be of type left-. Examples are provided showing that for each of the results all of the hypotheses are necessary.