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An inverse semigroup $S$ is a Howson inverse semigroup if the intersection offinitely generated inverse subsemigroups of $S$ is finitely generated. Given alocally finite action $\theta$ of a group $G$ on a semilattice $E$, it isproved that $E \ast_{\theta} G$ is a Howson inverse semigroup if and only if$G$ is a Howson group. It is also shown that this equivalence fails forarbitrary actions.

Howson's property for semidirect products of semilattices by groups