Advertisement

Pseudovarieties of Ordered Completely Regular Semigroups

  • Jorge AlmeidaEmail author
  • Ondřej Klíma
Article
  • 34 Downloads

Abstract

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role is played by the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.

Keywords

Ordered semigroup pseudovariety completely regular semigroup band complete lattice homomorphism 

Mathematics Subject Classification

20M19 20M07 20M35 

Notes

Acknowledgements

The authors would like to thank the referees for their comments and suggestions which contributed to improve the readability of the paper.

References

  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1995). English translationCrossRefGoogle Scholar
  2. 2.
    Almeida, J.: A syntactical proof of locality of DA. Int. J. Algebra Comput. 6, 165–177 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Almeida, J.: Profinite semigroups and applications. In: Kudryavtsev, V.B., Rosenberg, I.G. (eds.) Structural Theory of Automata, Semigroups and Universal Algebra, pp. 1–45. Springer, New York (2005)Google Scholar
  4. 4.
    Almeida, J., Klíma, O.: Representations of relatively free profinite semigroups, irreducibility, and order primitivity. Tech. rep., Univ. Masaryk and Porto (2015). ArXiv:1509.01389, submitted
  5. 5.
    Almeida, J., Klíma, O.: Towards a pseudoequational proof theory. Port. Math. 75, 79–119 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Almeida, J., Trotter, P.G.: Hyperdecidability of pseudovarieties of orthogroups. Glasg. Math. J. 43, 67–83 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Auinger, K., Hall, T.E., Reilly, N.R., Zhang, S.: Congruences on the lattice of pseudovarieties of finite semigroups. Int. J. Algebra Comput. 7, 433–455 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Birjukov, A.P.: Varieties of idempotent semigroups. Algebra Log. 9, 255–273 (1970)MathSciNetGoogle Scholar
  9. 9.
    Bloom, S.L.: Varieties of ordered algebras. J. Comput. Syst. Sci. 13, 200–212 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Emery, S.J.: Varieties and pseudovarieties of ordered normal bands. Semigr. Forum 58, 348–366 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fennemore, C.: All varieties of bands. I. Math. Nachr. 48, 237–252 (1971)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fennemore, C.: All varieties of bands. II. Math. Nachr. 48, 253–262 (1971)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gerhard, J.A.: The lattice of equational classes of idempotent semigroups. J. Algebra 15, 195–224 (1970)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Howie, J.M.: Fundamentals of semigroup theory. London Mathematical Society Monographs. New Series, vol. 12. The Clarendon Press, Oxford University Press, New York (1995)Google Scholar
  15. 15.
    Koch, R.J., Wallace, A.D.: Stability in semigroups. Duke Math. J. 24, 193–195 (1957)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Krohn, K., Rhodes, J.: Complexity of finite semigroups. Ann. Math. 2(88), 128–160 (1968)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kufleitner, M., Weil, P.: On \({\bf FO}^2\) quantifier alternation over words. In: Mathematical Foundations of Computer Science 2009. Lecture Notes in Computer Science, vol. 5734, pp. 513–524. Springer, Berlin (2009)Google Scholar
  18. 18.
    Kuřil, M.: On varieties of ordered semigroups. Semigr. Forum 90, 475–490 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Molchanov, V.A.: Nonstandard characterization of pseudovarieties. Algebra Univers. 33, 533–547 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pastijn, F.: Pseudovarieties of completely regular semigroups. Semigr. Forum 42, 1–46 (1991)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pastijn, F.J., Trotter, P.G.: Residual finiteness in completely regular semigroup varieties. Semigr. Forum 37, 127–147 (1988)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Petrich, M., Reilly, N.: Completely regular semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 23. Wiley-Interscience, New York (1999)Google Scholar
  23. 23.
    Pin, J.E.: A variety theorem without complementation. Russ. Math. (Iz. VUZ) 39, 80–90 (1995)Google Scholar
  24. 24.
    Pin, J.E., Weil, P.: A Reiterman theorem for pseudovarieties of finite first-order structures. Algebra Univers. 35, 577–595 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Polák, L.: On varieties of completely regular semigroups I. Semigr. Forum 32, 97–123 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Polák, L.: On varieties of completely regular semigroups II. Semigr. Forum 36, 253–284 (1987)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Polák, L.: On varieties of completely regular semigroups III. Semigr. Forum 37, 1–30 (1988)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Reilly, N.R., Zhang, S.: Complete endomorphisms of the lattice of pseudovarieties of finite semigroups. Bull. Aust. Math. Soc. 55, 207–218 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Reiterman, J.: The Birkhoff theorem for finite algebras. Algebra Univers. 14, 1–10 (1982)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rhodes, J., Steinberg, B.: The \(q\)-Theory of Finite Semigroups. Springer Monographs in Mathematics. Springer, Berlin (2009)zbMATHGoogle Scholar
  31. 31.
    Schützenberger, M.P.: Sur le produit de concaténation non ambigu. Semigr. Forum 13, 47–75 (1976)CrossRefGoogle Scholar
  32. 32.
    Tesson, P., Thérien, D.: Diamonds are forever: the variety DA. In: Gomes, G.M.S., Pin, J.E., Silva, P.V. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 475–499. World Scientific, Singapore (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CMUP, Department of Matemática, Faculdade de CiênciasUniversidade do PortoPortoPortugal
  2. 2.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations