Structure theorems for ordered groupoids

Abstract

The Ehresmann-Schein-Nambooripad theorem, which states that the category of inverse semigroups is isomorphic to the category of inductive groupoids, suggests a route for the generalisation of ideas from inverse semigroup theory to the more general setting of ordered groupoids. We use ordered groupoid analogues of the maximum group image and the E-unitary property – namely the level groupoid and incompressibility – to address structural questions about ordered groupoids. We extend the definition of the Margolis-Meakin graph expansion to an expansion of an ordered groupoid, and show that an ordered groupoid and its expansion have the same level groupoid and that the incompressibility of one determines the incompressibility of the other. We give a new proof of a P-theorem for incompressible ordered groupoids based on the Cayley graph of an ordered groupoid, and also use Ehresmann’s Maximum Enlargement Theorem to prove a generalisation of the P-theorem for more general immersions of ordered groupoids. We then carry out an explicit comparison between the Gomes-Szendrei approach to idempotent pure maps of inverse semigroups and our construction derived from the Maximum Enlargement Theorem.