Howson's property for semidirect products of semilattices by groups

TL;DR

This paper characterizes when semidirect products of semilattices by groups are Howson inverse semigroups, showing the property depends on the group being Howson, with implications for algebraic structure analysis.

Contribution

It establishes a precise condition linking the Howson property of the semidirect product to the group being Howson, for locally finite actions.

Findings

$E \ast_{\theta} G$ is Howson iff $G$ is Howson for locally finite actions

The equivalence does not hold for arbitrary actions

Provides criteria for the Howson property in inverse semigroup constructions

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 $\theta$ of a group $G$ on a semilattice $E$, it is proved 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 for arbitrary actions.