On connections between Morita semigroups and strong Morita equivalence

TL;DR

This paper explores the relationship between Morita semigroups and strong Morita equivalence, extending classical theorems to semigroups with weak local units and providing new characterizations.

Contribution

It establishes a connection between Morita semigroups and strong Morita equivalence for semigroups with weak local units, generalizing existing theorems.

Findings

Morita semigroups can be constructed via surjective Morita contexts.

Strong Morita equivalence characterized through Morita semigroups and isomorphisms.

Generalization of Hotzel's theorems to semigroups with weak local units.

Abstract

A surjective Morita context connecting semigroups $S$ and $T$ yields a Morita semigroup and a strict local isomorphism from it onto $S$ along which idempotents lift. We describe strong Morita equivalence of firm semigroups in terms of Morita semigroups and isomorphisms. We also generalize some of Hotzel's theorems to semigroups with weak local units. In particular, the Morita semigroup induced by a dual pair $\beta$ over a semigroup with weak local units is isomorphic to $\Sigma ^\beta$.