A 2-base for inverse semigroups

TL;DR

This paper proves that the identities x(x'x) = x and x (x' (y (y' ((z u)' w')'))) = y (y' (x (x' ((w z) u)))) form a 2-base for inverse semigroups, solving a long-standing open problem in algebra.

Contribution

The paper establishes a minimal 2-base for inverse semigroups, confirming a longstanding open problem and providing a simplified proof using automated deduction tools.

Findings

Identifies a 2-base for inverse semigroups.

Uses automated theorem proving to simplify the proof.

Provides a historical overview of the problem and its solution.

Abstract

An open problem in the theory of inverse semigroups was whether the variety of such semigroups, when viewed as algebras with a binary operation and a unary operation, is 2-based, that is, has a base for its identities consisting of 2 independent axioms. In this note, we announce the affirmative solution to this problem: the identities \[ \quad x(x'x) = x \qquad \quad x (x' (y (y' ((z u)' w')'))) = y (y' (x (x' ((w z) u)))) \] form a base for inverse semigroups where ${}'$ turns out to be the natural inverse operation. We recount here the history of the problem including our previous efforts to find a 2-base using automated deduction and the method that finally worked. We describe our efforts to simplify the proof using \textsc{Prover9}, present the simplified proof itself and conclude with some open problems.