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.Caledonian Research Foundation Scholarshi
