  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.
