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.
This is a preview of subscription content, log in to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
S.M. Armstrong, Structure of concordant semigroups. J. Algebra 118, 205–260 (1988)
N. AlYamani, N.D. Gilbert, E.C. Miller, Fibrations of ordered groupoids and the factorization of ordered functors. Appl. Categ. Struct. 24, 121–146 (2016)
M.J.J. Brancoa, G.M.S. Gomes, V. Gould, Ehresmann monoids. J. Algebra 443, 349–382 (2015)
C. Cornock, V. Gould, Proper two-sided restriction semigroups and partial actions. J. Pure Appl. Algebra 216, 935–949 (2012)
D. Dewolf, D. Pronk, The Ehresmann–Schein–Nambooripad theorem for inverse categories. Theory Appl. Categ. 33, 813–831 (2018)
J.B. Fountain, G.M.S. Gomes, V. Gould, A Munn type representation for a class of \(E\)-semiadequate semigroups. J. Algebra 218, 693–714 (1999)
V. Gould, Notes on restriction semigroups and related structures. (2010). https://www.researchgate.net/publication/237604491
V. Gould, Y.H. Wang, Beyond orthodox semigroups. J. Algebra 368, 209–230 (2012)
V. Gould, Restriction and Ehresmann semigroups, in ed. by H. Wanida, W. Sri, W.S. Polly, Proceedings of the International Conference on Algebra 2010: Advances in Algebraic Structures, (World Sci. Publ., Hackensack, 2012), pp. 265–288
V. Gould, M.B. Szendrei, Proper restriction semigroups–semidirect products and \(W\)-products. Acta Math. Hung. 141, 36–57 (2013)
V. Gould, M. Hartmann, M.B. Szendrei, Embedding in factorisable restriction monoids. J. Algebra 476, 216–237 (2017)
C. Hollings, From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2, 21–57 (2009)
C. Hollings, Extending Ehresmann–Schein–Nambooripad theorem. Semigroup Forum 80, 453–476 (2010)
C. Hollings, The Ehresmann–Schein–Nambooripad theorem and its successors. Eur. J. Pure Appl. Math. 5, 414–450 (2012)
J.M. Howie, An introduction to semigroup theory (Academic Press, London, 1976)
M. Jackson, T. Stokes, An invitation to \(C\)-semigroups. Semigroup Forum 62, 279–310 (2001)
P.R. Jones, A common framework for restriction semigroups and regular \(\ast \)-semigroups. J. Pure Appl. Algebra 216, 618–632 (2012)
P.R. Jones, Varieties of restriction semigroups and varieties of categories. Commun. Algebra 45, 1037–1056 (2017)
G. Kudryavtseva, Partial monoid actions and a class of restriction semigroups. J. Algebra 429, 342–370 (2015)
M.V. Lawson, Semigroups and ordered categories. I. The reduced case. J. Algebra 141, 422–462 (1991)
M.V. Lawson, Inverse Semigroups, the Theory of Partial Symmetries (World Scientific, Singapore, 1998)
J. Meakin, The structure mappings of a regular semigroup. Proc. Edinb. Math. Soc. 21, 135–142 (1978)
K.S.S. Nambooripad, Structure of regular semigroups. Mem. Amer. Math. Soc. 22(224), 1–117 (1979)
T. Nordahl, H.E. Scheiblich, Regular \(\ast \)-semigroups. Semigroup Forum 16, 369–377 (1978)
M.B. Szendrei, Embedding of a restriction semigroup into a \(W\)-product. Semigroup Forum 89, 280–291 (2014)
Y.H. Wang, Beyond regular semigroups. Semigroup Forum 92, 414–448 (2016)
S.F. Wang, Fundamental regular semigroups with quasi-ideal regular \(\ast \)-transversals. Bull. Malays. Math. Sci. Soc. 38, 1067–1083 (2015)
S.F. Wang, On algebras of \(P\)-Ehresmann semigroups and their associate partial semigroups. Semigroup Forum 95, 569–588 (2017)
S.F. Wang, An Ehresmann–Schein–Nambooripad-type theorem for a class of \(P\)-restriction demigroups. Bull. Malays. Math. Sci. Soc. 42, 535–568 (2019)
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  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).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
- Locally Ehresmann P-Ehresmann semigroup
- Lepe-generalized category
- Projection algebra
- ESN theorem
Mathematics Subject Classification