## 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.

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## Acknowledgements

The author expresses his profound gratitude to the referee for the valuable comments and suggestions, which improve greatly the content and presentation of this article. According to the referee’s advices, we add a new section to state the main results in [29] and make some connections with the results in the present paper. Thanks also go to Professor Maria B. Szendrei for the timely communications. This paper is supported by Nature Science Foundation of China (11661082).

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Wang, S. An Ehresmann–Schein–Nambooripad theorem for locally Ehresmann *P*-Ehresmann semigroups.
*Period Math Hung* (2020). https://doi.org/10.1007/s10998-019-00309-x

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### Keywords

- Locally Ehresmann
*P*-Ehresmann semigroup - Lepe-generalized category
- Projection algebra
- ESN theorem

### Mathematics Subject Classification

- 20M10