An Ehresmann–Schein–Nambooripad theorem for locally Ehresmann P-Ehresmann semigroups

Abstract

Lawson has obtained an Ehresmann–Schein–Nambooripad theorem (ESN theorem for short) for Ehresmann semigroups which states that the category of Ehresmann semigroups together with (2,1,1)-homomorphisms is isomorphic to the category of Ehresmann categories together with admissible mappings. Recently, Jones introduced P-Ehresmann semigroups as generalizations of Ehresmann semigroups. In this paper, we shall study P-Ehresmann semigroups by “category approach”. In spirit of Lawson’s methods, we introduce the notion of lepe-generalized categories by which locally Ehresmann P-Ehresmann semigroups are described. Moreover, we show that the category of locally Ehresmann P-Ehresmann semigroups together with (2,1,1)-homomorphisms is isomorphic to the category of lepe-generalized categories over local semilattices together with admissible mappings. Our work may be regarded as extending the ESN theorem for Ehresmann semigroups. Some special cases are also considered.