Sridhar Ramesh: Is there meant to also be a partial ordering on the objects in addition to those on the Hom-sets? (Without this, I cannot make sense of the source and target maps preserving the partial order. Indeed, as it stands, I don’t see how this definition is any different from that of a locally partially ordered category.)
David Roberts: I agree: the first definition is of a category enriched in Pos, whereas the reference to source and target maps clearly talks about an internal category. I’ve edited it.
Partially ordered -categories
For a partially ordered -category we can (optionally?) require that
for any morphisms and .
The following nomenclature points were raised by one of the previous contruibutors. What do people think? Are they justified or in conflict with existing terminology? Are there examples where the conditions come up naturally?
For a partially ordered -category I will call monovalued morphism such a morphism that .
For a partially ordered category with inverses I will call entirely defined morphism such a morphism that .
Partially ordered groupoids
The particular case of an ordered groupoid (in which each is the inverse of ) is called an ordered groupoid. This has been studied extensively by Mark Lawson, for instance see
Lawson, Mark V Constructing ordered groupoids Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123-138, (numdam)
or his book:
M.V. Lawson, Inverse semigroups: the theory of partial symmetries , World Scientific, 1998.
He makes the point that they were initially studied by Ehresmann and are very closely related to inverse semi-groups.