Abstract
The aim of this paper is to study weakly \(U\)-regular semigroups, a wide class containing all regular semigroups and all abundant semigroups with a regular biordered set of idempotents. Here \(U\) is a regular biordered set. To do this, we introduce the notions of an RBS category and a weakly regular category over a regular biordered set. We show that the category of weakly \(U\)-regular semigroups and admissible morphisms is equivalent to the category of weakly regular categories and RBS functors. Our method arises from Nambooripad’s work on the connection between regular biordered sets and regular semigroups. However, there are completely different techniques, the first being the introduction of RBS categories and the second being that it is more convenient to investigate (RBS) categories equipped with pre-orders, rather than with partial orders. A special case of our work is the class of weakly \(U\)-orthodox semigroups, that is, weakly \(U\)-regular semigroups with \(U\) a band, characterised in an earlier article by the author using generalised categories equipped with pre-orders. Our result may be regarded as an extension of Armstrong’s work on concordant semigroups in the abundant case.
Similar content being viewed by others
References
Armstrong, S.M.: Structure of concordant semigroups. J. Algebra 118, 205–260 (1988)
Ehresmann, C.: ‘Oeuvres complètes et comentées’. Suppl. Cahiers Topologie Géom. Différenti\(\grave{e}\)lle (Amiens, 1980–1984)
Gould, V.A.R., Wang, Y.H.: Beyond orthodox semigroups. J. Algebra 368, 209–230 (2012)
Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)
Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, New York (1976)
Kilp, M., Knauer, U., Mikhalev, A.V.: Monoids, acts and categories: with applications to wreath products and graphs: a handbook for students and researchers. In: De Gruyter Expositions in Mathematics, vol. 29. Walter de Gruyter, Berlin (2000)
Lawson, M.V.: Semigroups and ordered categories. I. the reduced case. J. Algebra 141, 422–462 (1991)
Nambooripad, K.S.S.: Structure of regular semigroups, I. Mem. Amer. Math. Soc. 22(224) (1979)
Nambooripad, K.S.S., Veeramony, R.: Subdirect products of regular semigroups. Semigroup Forum 28, 265–307 (1983)
Ren, X.M., Shum, K.P.: On superabundant semigroups whose set of idempotents forms a subsemigroup. Algebra Colloq. 14, 215–228 (2007)
Ren, X.M., Wang, Y.H., Shum, K.P.: On \(U\)-orthodox semigroups. Sci. China Ser. A 52, 329–350 (2009)
Ren, X.M., Yin, Q.Y., Shum, K.P.: On \(U^{\sigma }\)-abundant semigroups. Algebra Colloq. 19, 41–52 (2012)
Ren, X.M., Yang, D.D., Shum, K.P.: On locally Ehresmann semigroups. J. Algebra Appl. 10, 1165–1186 (2011)
Schein, B.M.: On the theory of inverse semigroups and generalised groups. Am. Math. Soc. Transl. Ser. 2(113), 89–112 (1979)
Simmons, C.P.: Small Category Theory Applied to Semigroups and Monoids, PhD Thesis, The University of York (2001)
Wang, Y.H.: Weakly B-orthodox semigroups. Period. Math. Hung. 68, 13–38 (2014)
Wang, Y.H.: Beyond Regular Semigroups, PhD Thesis, The University of York (2012)
Acknowledgments
The author was supported by the National Natural Science Foundation of China (Grant No:11471255), the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and SDUST Research Fund (No. 2014TDJH102); Joint Innovative Center for Safe And Effective Mining Technology and Equipment of Coal Resources, Shandong Province. This work is a part of the author’s PhD thesis. She would like to take this opportunity to thank charitable sponsors in Hong Kong for supporting her to study in the University of York. She would like to thank Prof. Victoria Gould for carefully reading and revising her thesis and for providing useful suggestions. She should like to thank Prof. K.P. Shum and Prof. X.M. Ren for, among many things, their greatest help and encouragement all the time.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by László Márki.
Rights and permissions
About this article
Cite this article
Wang, Y. Beyond regular semigroups. Semigroup Forum 92, 414–448 (2016). https://doi.org/10.1007/s00233-015-9714-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-015-9714-4