Generalized heaps, inverse semigroups and Morita equivalence

Inverse semigroups are the algebraic counterparts of pseudogroups of transformations. The algebraic 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.