Advertisement

Semigroup Forum

, Volume 97, Issue 2, pp 307–324 | Cite as

Restriction \(\omega \)-semigroups

  • Yanhui WangEmail author
  • Dilshad Abdulkadir
RESEARCH ARTICLE
  • 91 Downloads

Abstract

The purpose of this paper is to investigate restriction \(\omega \)-semigroups. Here a restriction \(\omega \)-semigroup is a generalisation of an inverse \(\omega \)-semigroup. We give a description of a class of restriction \(\omega \)-semigroups, namely, restriction \(\omega \)-semigroups with an inverse skeleton. We show that a restriction \(\omega \)-semigroup with an inverse skeleton is an ideal extension of a \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse \(\omega \)-semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup with an inverse skeleton, conversely, every \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroup with an inverse skeleton arises in this way.

Keywords

Bruck–Reilly semigroups \(\widetilde{\mathcal {J}}\)-simple restriction \(\omega \)-semigroups Inverse skeleton 

Notes

Acknowledgements

We would like to thank the referee for some suggestions about the definition of restriction semigroups, the literary background and Example 5.1. The authors would also like to thank Victoria Gould for her suggestions and comments on their manuscript.

References

  1. 1.
    Abdulkadir, D.: \(\widetilde{\cal{J}}\)-restriction \(\omega \)-semigroups. Ph.D. thesis, The University of York (2014)Google Scholar
  2. 2.
    Asibong-Ibe, U.: \(\ast \)-bisimple type A \(\omega \)-semigroups-I. Semigroup Forum 31, 99–117 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asibong-Ibe, U.: \(\ast \)-simple type A \(\omega \)-semigroups. Semigroup Forum 47, 135–149 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Clifford, A.H.: A class of \(d\)-simple semigroups. Am. J. Math. 75, 547–556 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fountain, J.: Adequate semigroups. Proc. Edinb. Math. Soc. 22, 113–125 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fountain, J., Gomes, G.M.S., Gould, V.: A Munn type representation for a class of \(E\)-semiadequate semigroups. J. Algebra 218, 693–714 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gould, V.: Notes on restriction semigroups and related structures 2017. http://www-users.york.ac.uk/~varg1/restriction.pdf
  8. 8.
    Gould, V., Hollings, C.: Restriction semigroups and inductive constellations. Commun. Algebra 38, 261–287 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gould, V., Zenab, R.: Semigroups with inverse skeletons and Zappa–Szep products. CGASA 1, 59–89 (2013)zbMATHGoogle Scholar
  10. 10.
    Howie, J.M.: Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series Volume 12. The Clarendon Press, New York (1995)Google Scholar
  11. 11.
    Jackson, M., Stokes, T.: An invitation to \(C\)-semigroups. Semigroup Forum 62, 279–310 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jones, P.R.: On lattices of varieties of restriction semigroups. Semigroup Forum 86, 337–361 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jones, P.R.: The semigroups \(B_2\) and \(B_0\) are inherently nonfinitely based, as restriction semigroups. Int. J. Algebra Comput. 23, 1289–1335 (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Jones, P.R.: Almost perfect restriction semigroups. J. Algebra 445, 193–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kochin, B.P.: The structure of inverse ideally-simple \(\omega \)-semigroups. Vestnik Leningradskogo Universiteta 23, 41–50 (1968). (Russian)Google Scholar
  16. 16.
    Lawson, M.V.: Inverse Semigroups. World Scientific Publishing Co. Pte. Ltd, Singapore (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Munn, W.D.: Regular \(\omega \)-semigroups. Glasg. Math. J. 9, 46–66 (1968)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ma, S., Ren, X., Yuan, Y.: On \(U\)-ample \(\omega \)-semigroups. Front. Math. China 8, 1391–1405 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rees, D.: On the ideal structure of a semigroup satisfying a cancellation law. Q. J. Math. 19, 101–108 (1948)CrossRefzbMATHGoogle Scholar
  20. 20.
    Reilly, N.R.: Bisimple inverse \(\omega \)-semigroups. Proc. Glasg. Math. Assoc. 7, 160–167 (1966)CrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, Y.H.: Weakly B-orthodox semigroups. Period. Math. Hung. (68) 1, 13–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, Y.H.: Hall-type representations for generalised orthogroups. Semigroup Forum (89) 3, 518–545 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, Y.H.: Beyond regular semigroups. Semigroup Forum (92) 2, 414–448 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.LeedsUK

Personalised recommendations