Skip to main content
SpringerLink
Log in

Some relations on a semilattice of varieties of completely regular semigroups

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Denote by \(\mathcal {CR}\) the variety of all completely regular semigroups with the unary operation of inversion within their maximal subgroups. Next let \(\mathcal {L}^*\) be the subvariety of \(\mathcal {CR}\) in all of whose members Green’s relation \(\mathcal {L}\) is a congruence. For an \(\cap \)-subsemilattice of subvarieties of \(\mathcal {L}^*\), we determine the lower ends of kernel, (left and right) trace, and core relations, and a basis of identities of its members with a few exceptions. The semilattice is presented by a diagram, and the results in a table.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Diagram 1
Diagram 2
Diagram 3
Diagram 4

Similar content being viewed by others

References

  1. Jones, P.R.: Mal’cev products of varieties of completely regular semigroups. J. Aust. Math. Soc. A 42, 227–246 (1987)

    Article  MATH  Google Scholar 

  2. Kaďourek, J.: On the word problem for free bands of groups and for free objects in some other varieties of completely regular semigroups. Semigroup Forum 38, 1–55 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Pastijn, F.: The lattice of completely regular semigroup varieties. J. Aust. Math. Soc. Ser. A 49, 24–42 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Pastijn, F., Yan, X.: Varieties of semigroups and varieties of completely regular semigroups closed for certain extensions. J. Algebra 163, 777–794 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Petrich, M.: On the varieties of completely regular semigroups. Semigroup Forum 25, 153–169 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Petrich, M.: Canonical varieties of completely regular semigroups. J. Aust. Math. Soc. 83, 87–104 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Petrich, M.: Embedding regular semigroups into idempotent generated ones. Algebra Colloq. 17, 229–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Petrich, M.: A lattice of varieties of completely regular semigroups. Commun. Algebra 42, 1397–1413 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Petrich, M.: Varieties of completely regular semigroups related to canonical varieties. Semigroup Forum 90, 53–99 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Petrich, M.: Certain relations on a lattice of varieties of completely regular semigroups. Commun. Algebra 43, 4080–4096 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Petrich, M., Reilly, N.R.: Semigroups generated by certain operators on varieties of completely regular semigroups. Pac. J. Math. 132, 151–175 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Petrich, M., Reilly, N.R.: Operators related to \(E\)-disjunctive and fundamental completely regular semigroups. J. Algebra 134, 1–27 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Petrich, M., Reilly, N.R.: Operators related to idempotent generated and monoid completely regular semigroups. J. Aust. Math. Soc. Ser. A 49, 1–23 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Petrich, M., Reilly, N.R.: Completely Regular Semigroups. Wiley, New York (1999)

    MATH  Google Scholar 

  15. Polák, L.: On varieties of completely regular semigroups I. Semigroup Forum 32, 97–123 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Polák, L.: On varieties of completely regular semigroups II. Semigroup Forum 36, 253–284 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Assistance of Edmond Lee is deeply appreciated.

Author information

Authors and Affiliations

  1. 21420, Bol, Brač, Croatia

    Mario Petrich

Authors
  1. Mario Petrich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Petrich.

Additional information

Communicated by László Márki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petrich, M. Some relations on a semilattice of varieties of completely regular semigroups. Semigroup Forum 93, 607–628 (2016). https://doi.org/10.1007/s00233-016-9817-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-016-9817-6

Keywords

Mathematics Subject Classification

Search

Navigation