Inverse semigroups of left I-quotients

TL;DR

This paper introduces a general framework for inverse semigroups of left quotients, called left I-quotients, extending Clifford's classical work and clarifying the structure of bisimple inverse semigroups.

Contribution

It develops a new, unified approach to inverse semigroups of left quotients, simplifying the extension of Clifford's results to bisimple inverse semigroups.

Findings

Extended Clifford's work to bisimple inverse semigroups

Provided a clearer framework for Gantos' earlier work

Facilitated further research in inverse semigroup theory

Abstract

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford's seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford's work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work on Gantos into a wider and clearer context, and pave the way for further progress.