Generalized heaps, inverse semigroups and Morita equivalence

TL;DR

This paper establishes a bijective correspondence between generalized heaps and equivalence bimodules, demonstrating that Morita equivalence of inverse semigroups can be characterized through Wagner's generalized heaps, linking algebraic and geometric structures.

Contribution

It proves that generalized heaps precisely correspond to equivalence bimodules, providing a new perspective on Morita equivalence of inverse semigroups.

Findings

Generalized heaps are in bijection with equivalence bimodules.

Morita equivalence of inverse semigroups is characterized by generalized heaps.

The work links algebraic and geometric frameworks in inverse semigroup theory.

Abstract

Inverse semigroups are the abstract counterparts of pseudogroups of transformations. The abstract counterparts of atlases in differential geometry are what Wagner termed `generalized heaps'. These are sets equipped with a ternary operation satisfying certain axioms. We prove that there is a bijective correspondence between generalized heaps and the equivalence bimodules, defined by Steinberg. Such equivalence bimodules are used to define the Morita equivalence of inverse semigroups. This paper therefore shows that the Morita equivalence of inverse semigroups is determined by Wagner's generalized heaps.