Finiteness Conditions for Clifford Semigroups

Abstract

There is a large amount of published work in the last decade on finiteness conditionsof monoids and groups such as n FP and its siblings. Recently Gray and Pride havefound that a Clifford monoid containing a minimal idempotent e is of type n FP ifand only if its maximal subgroup containing e is of the same type. In our paper welook for results which are in the same spirit as the above, that is, we try to relate thehomological finiteness conditions of a Clifford monoid to those of a certain grouparising from its semilattice structure. More specifically, we prove that if acommutative Clifford monoid S is of type n FP , then its maximum group image Gis of the same type. To achieve this we employ a result of [10] which relates thecohomology groups of S to those of G, and the fact that the functor n ( , )S Ext   commutes with direct limits whenever S is of type n FP .