Morita equivalence of semigroups with local units

TL;DR

This paper characterizes Morita equivalence of semigroups with local units through joint enlargements, linking it to the structure of regular and inverse semigroups, and clarifies their local structural relationships.

Contribution

It establishes a new criterion for Morita equivalence of semigroups with local units using joint enlargements, connecting it to inverse semigroups and local inverse semigroups.

Findings

Morita equivalence characterized by joint enlargements

Semigroup with local units is Morita equivalent to an inverse semigroup iff it is regular locally inverse

Provides a framework linking Morita theory to McAlister's local structure theory

Abstract

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister's theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.