A note on the Howson property in inverse semigroups

TL;DR

This paper investigates the conditions under which inverse semigroups possess the Howson property, providing a characterization for those with finitely many idempotents and establishing that all monogenic inverse semigroups have this property.

Contribution

It offers a necessary and sufficient condition for inverse semigroups with finitely many idempotents to have the Howson property and proves that all monogenic inverse semigroups do.

Findings

Inverse semigroups with finitely many idempotents satisfy the Howson property under a specific condition.

All monogenic inverse semigroups possess the Howson property.

Provides a characterization linking idempotent count to the Howson property.

Abstract

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.