Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the `join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the `morphisms' part. However, it is so-called `meet-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for meet-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.Comment: 23 pages; final section on Szendrei expansions removed; further reordering of materia

On Ehresmann semigroups

We describe an alternative approach to describing Ehresmann semigroups by categories in which a class of \'etale actions plays an important r\^ole. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As an application, we prove that every birestriction semigroup can be suitably embedded into a birestriction semigroup constructed from a category. As a corollary, we determine when a birestriction semigroup can be suitably embedded into an inverse semigroup