An elegant 3-basis for inverse semigroups

TL;DR

This paper proves that a specific algebraic structure satisfying three identities is equivalent to an inverse semigroup, providing a new characterization of inverse semigroups with a simplified basis.

Contribution

It establishes that the three identities are sufficient to characterize inverse semigroups, offering a new, elegant basis for their algebraic description.

Findings

The three identities characterize inverse semigroups.

Unary operation coincides with the usual inversion.

Provides a simplified basis for inverse semigroup theory.

Abstract

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: [\quad x=(xx')x \qquad \quad (xx')(y'y)=(y'y)(xx') \qquad \quad (xy)z=x(yz"). ] The goal of this note is to prove the converse, that is, we prove that an algebra of type $<2,1>$ satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.