Howson's Property for Semidirect Products of Semilattices by Groups

Abstract

An inverse semigroup S is a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups of S is finitely generated. Given a locally finite action of a group G on a semilattice E, it is proved that E*G is a Howson inverse semigroup if and only if G is a Howson group. It is also shown that this equivalence fails for arbitrary actions.