Homological finiteness properties of monoids, their ideals and maximal subgroups

Details Homological finiteness properties of monoids, their ideals and maximal subgroups Homological finiteness properties of monoids, their ideals and maximal subgroups

2010-03-16

arxiv.org/abs/1003.3227v1

Mathematics

Group Theory

25 pages

Scientific paper

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.

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