A locally partially ordered category is a category together with partial order on each of its hom-sets such that
for any morphisms , , , such that the composites above are defined. In other words, a category enriched in Pos, the category of posets.
Compare this to the notion of category internal to Pos.
Similarly, such a partially ordered category can be considered as a special kind of double category, while a locally partially ordered category can be considered as a special kind of -category.
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 .
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
(numdam)
or his book:
He makes the point that they were initially studied by Ehresmann and are very closely related to inverse semi-groups.
Last revised on September 5, 2022 at 13:53:41. See the history of this page for a list of all contributions to it.