Effective equivalence relations and principal quantales

TL;DR

This paper introduces principal quantales as a non-unital generalization of supported quantales, linking algebraic structures to open groupoids and effective equivalence relations.

Contribution

It extends the theory of supported quantales to a non-unital setting, connecting algebraic structures with open groupoids and effective equivalence relations.

Findings

Principal quantales correspond to effective equivalence relations in groupoid theory.

Generalization from unital to non-unital supported quantales broadens the algebraic framework.

Provides a new algebraic perspective on open groupoids and their relations.

Abstract

Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and \'etale groupoids. Here we study a further generalization where a non-unital version of supported quantale carries the algebraic content of such correspondences to the setting of open groupoids. A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.