Anisimov's Theorem for inverse semigroups

TL;DR

This paper proves that finitely generated inverse semigroups with regular idempotent problems are finite, extending Anisimov's Theorem from groups to inverse semigroups and answering a longstanding question.

Contribution

It establishes a generalization of Anisimov's Theorem to inverse semigroups, showing the finiteness of those with regular idempotent problems.

Findings

Finitely generated inverse semigroups with regular idempotent problems are finite.

Answers a question posed by Gilbert and Noonan Heale.

Extends Anisimov's Theorem from groups to inverse semigroups.

Abstract

The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalisation to inverse semigroups of Anisimov's Theorem for groups.