Generalized restriction P-restriction semigroups are common generalizations of restriction semigroups and generalized inverse \(*\)-semigroups. Gomes and Szendrei (resp. Ohta and Imaoka) have shown that every restriction semigroup (every generalized inverse \(*\)-semigroup) can be embedded in a complete, infinitely distributive restriction semigroup (resp. a \(*\)-complete, infinitely distributive generalized inverse \(*\)-semigroup). The main aim of this paper is to obtain an entirely corresponding result for generalized restriction P-restriction semigroups. Specifically, among other things, we show that every generalized restriction P-restriction semigroup can be (2,1,1)-embedded in a complete, infinitely distributive generalized restriction P-restriction semigroup. Our results generalize and enrich the corresponding results of Gomes, Szendrei, Ohta and Imaoka.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Auinger, K.: Free locally inverse \(\ast \)-semigroup. Czechoslov. Math. J. 43, 523–545 (1993)
Burgess, W.D.: Completions of semilattices of cancellative semigroups. Glasg. Math. J. 21, 29–37 (1980)
Gomes, G.M.S., Szendrei, M.B.: Almost factorizable weakly ample semigroups. Commun. Algebra 35, 3503–3523 (2007)
Gould, V.: Restriction and Ehresmann semigroups. In: Proceedings of the International Conference on Algebra (2010), pp. 265–288. World Sci. Publ., Hackensack, NJ (2012)
Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, London (1976)
Hall, T.E., Imaoka, T.: Representations and amalgamation of generalized inverse \(\ast \)-semigroups. Semigroup Forum 58, 126–141 (1999)
Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2, 21–57 (2009)
Imaoka, T.: On fundamental regular \(\ast \)-semigroups. Mem. Fac. Sci. Shimane Univ. 14, 19–23 (1980)
Imaoka, T.: Representations of generalized inverse \(\ast \)-semigroups. Acta Sci. Math. (Szeged) 61, 171–180 (1995)
Jones, P.R.: A common framework for restriction semigroups and regular \(\ast \)-semigroups. J. Pure Appl. Algebra 216, 618–632 (2012)
Jones, P.R.: Varieties of \(P\)-restriction semigroups. Commun. Algebra 42, 1811–1834 (2014)
Jones, P.R.: Almost perfect restriction semigroups. J. Algebra 445, 193–220 (2016)
Kudryavtseva, G.: Partial monoid actions and a class of restriction semigroups. J. Algebra 429, 342–370 (2015)
Lawson, M.V.: Covering and embeddings of inverse semigroups. Proc. Edinb. Math. Soc. 36, 399–419 (1993)
Lawson, M.V.: Almost factorizable inverse semigroups. Glasg. Math. J. 36, 97–111 (1994)
Lawson, M.V.: Inverse Semigroups. World Scientific, Singapre (1998)
Leech, J.: Inverse monoids with a natural semilattice ordering. Proc. Lond. Math. Soc. 70, 146–182 (1995)
McAlister, D.B., Reilly, N.R.: E-unitary covers for inverse semigroups. Pac. J. Math. 68, 161–174 (1977)
Nordahl, T.E., Scheiblich, H.E.: Regular \(\ast \)-semigroups. Semigroup Forum 16, 369–377 (1978)
Ohta, H., Imaoka, T.: Completions of generalized inverse \(\ast \)-semigroups. RIMS Kokyuroku 1604, 114–119 (2008)
Petrich, M.: Inverse Semigroups. Wiley, New York (1984)
Pastijn, F.J., Oliveira, L.: Maximal dense ideal extensions of locally inverse semigroups. Semigroup Forum 72, 441–458 (2006)
Qallali, A.E., Fountain, J.: Proper covers for left ample semigroups. Semigroup Forum 71, 411–427 (2005)
Schein, B.M.: Completions, translational hulls and ideal extensions of inverse semigroups. Czechoslov. Math. J. 23, 575–610 (1973)
Scheiblich, H.E.: Generalized inverse semigroups with involution. Rocky Mt. J. Math. 12, 205–211 (1982)
Szendrei, M.B.: Free \(\ast \)-orthodox semigroups. Simon Stevin 59, 175–201 (1985)
Shoji, K.: Completions and injective hulls of \(E\)-reflexive inverse semigroups. Semigroup Forum 36, 55–68 (1987)
Szendrei, M.B.: Embedding into almost left factorizable restriction semigroups. Commun. Algebra 41, 1458–1483 (2013)
Wang, S.F.: On algebras of \(P\)-Ehresmann semigroups and their associate partial semigroups. Semigroup Forum 95, 569–588 (2017)
Wang, S.F.: An Ehresmann–Schein–Nambooripad-type theorem for a class of \(P\)-restriction semigroups. Bull. Malays. Math. Sci. Soc. 42, 535–568 (2019)
Wang, S.F.: An Ehresmann–Schein–Nambooripad theorem for locally Ehresmann \(P\)-Ehresmann semigroups. Periodica Mathematica Hungarica, to appear
The authors express their profound gratitude to the referee for the valuable comments and suggestions which not only improve the present paper but also give some new methods and thoughts for the future study. In particular, the referee has pointed out that some results of Sect. 3 can be deduced by using the fact that every generalized restriction P-restriction semigroup is a subdirect product of a restriction semigroup and a full (2,1,1)-subsemigroup of a generalized inverse \(*\)-semigroup. (This fact is essentially given by Jones in Proposition 5.5 of .) Thanks also go to the editor for the timely communications. This research is supported partially by Nature Science Foundation of China (11661082) and the Postgraduate Students Research Innovation Fund of Yunnan Normal University (2019).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Peyman Niroomand.
About this article
Cite this article
Yan, P., Wang, S. Completions of Generalized Restriction P-Restriction Semigroups. Bull. Malays. Math. Sci. Soc. 43, 3651–3673 (2020). https://doi.org/10.1007/s40840-020-00888-w
- Generalized restriction P-restriction semigroup
- Permissible set
Mathematics Subject Classification