Anisimov's Theorem for inverse semigroups

The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This paper 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 generalization to inverse semigroups of Anisimov's Theorem for groups.