Semigroup Forum

, Volume 97, Issue 3, pp 548–561 | Cite as

Regular semigroups with weakly simplistic orthodox transversals

  • Xiangjun KongEmail author
Research Article


Weakly simplistic orthodox transversals are introduced in this paper and some characterizations associated with them are obtained. An example is given to demonstrate that weakly simplistic orthodox transversals are proper generalisations of left simplistic orthodox transversals. The related results of Blyth and Almeida Santos on left simplistic inverse transversals obtained in 1996 and Kong and Luo on left simplistic orthodox transversals obtained in 2011 are generalised and amplified. A structure theorem of regular semigroups with weakly simplistic orthodox transversals is also established.


Regular semigroup Orthodox semigroup Weakly simplistic orthodox transversal 



The author would like to express his sincere thanks to M.Jackson and the referee for their valuable suggestions and corrections, which much improved this paper. The author is a postdoctoral researcher of the Postdoctoral Station of Qufu Normal University. The research is partially supported by NSF of Shandong Province (ZR2016AM02), the NSFC (11471186), A Project of Shandong Province Higher Educational Science and Technology Program (J18KA248) and Scientific Research Foundation of Qufu Normal University (xkj201509).


  1. 1.
    Blyth, T.S., Almeida Santos, M.H.: A simplistic approach to inverse transversals. Proc. Edinb. Math. Soc. 39, 57–69 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blyth, T.S., Almeida Santos, M.H.: \(\cal{H}\)-cohesive orders associated with inverse transversals. Commun. Algebra 40(8), 2771–2785 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blyth, T.S., McFadden, R.B.: Regular semigroups with a multiplicative inverse transversal. Proc. Roy. Soc. Edinburgh 92A, 253–270 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, J.F.: On regular semigroups with orthodox transversals. Commun. Algebra 27, 4275–4288 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, J.F.: Abundant semigroups with adequate transversals. Semigroup Forum 60, 67–79 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, J.F., Guo, Y.Q.: Orthodox transversals of regular semigroups. Int. J. Algebra Comput. 11(2), 269–279 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    El-Qallali, A.: Abundant semigroups with a multiplicative type A transversal. Semigroup Forum 47(3), 327–340 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  9. 9.
    Kong, X.J.: Regular semigroups with quasi-ideal orthodox transversals. Semigroup Forum 74, 247–258 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kong, X.J.: Some properties associated with adequate transversals. Can. Math. Bull. 54(3), 487–497 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kong, X.J.: On generalized orthodox transversals. Commun. Algebra 42(4), 1431–1447 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kong, X.J., Luo, Y.F.: Regular semigroups with left simplistic orthodox transversals. Sci. Sin. Math. 41(8), 745–757 (2011). (in Chinese) CrossRefGoogle Scholar
  13. 13.
    Kong, X.J., Meng, F.W.: The generalization of two basic results for orthodox semigroups. Semigroup Forum 89(2), 394–402 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kong, X.J., Wang, P.: The product of quasi-ideal adequate transversals of an abundant semigroup. Semigroup Forum 83(2), 304–312 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kong, X.J., Zhao, X.: A new construction for regular semigroups with quasi-ideal orthodox transversals. J. Aust. Math. Soc. 86, 177–187 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    McAlister, D.B., McFadden, R.B.: Regular semigroups with inverse transversals. Q. J. Math. Oxf. 34(2), 459–474 (1983)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Saito, T.: Construction of regular semigroups with inverse transversals. Proc. Edinb. Math. Soc. 32, 41–51 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tang, X.L.: Regular semigoups with inverse transversals. Semigroup Forum 55, 24–32 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuChina

Personalised recommendations