Glasnik Matematicki, Vol. 50, No. 2 (2015), 373-396.

THE VARIETY GENERATED BY ALL MONOIDS OF ORDER FOUR IS FINITELY BASED

Edmond W. H. Lee and Jian Rong Li

Department of Mathematics, Nova Southeastern University, Fort Lauderdale, Florida 33314, USA
e-mail: edmond.lee@nova.edu

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China
e-mail: lijr@lzu.edu.cn

Dedicated to Professor Mikhail V. Volkov on the occasion of his 60th birthday

Abstract.   It is known that the variety Mn generated by all monoids of order n is finitely based if n ≤ 3 and non-finitely based if n ≥ 6. The present article establishes the finite basis property of the variety M4. This leaves M5 as the last open case in the finite basis problem for the varieties Mn.

2010 Mathematics Subject Classification.   20M07.

Key words and phrases.   Monoid, semigroup, variety, finitely based.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.50.2.08


References:

  1. J. Almeida, Finite semigroups and universal algebra, World Scientific, Singapore, 1994.
    MathSciNet    

  2. A. P. Birjukov, Varieties of idempotent semigroups, Algebra i Logika 9 (1970), 255-273 (in Russian); English transl.: Algebra and Logic 9 (1970), 153-164.
    MathSciNet    

  3. A. Distler and J. D. Mitchell, Smallsemi - a GAP package, version 0.6.6, 2013, available at http://www.gap-system.org/Packages/smallsemi.html

  4. C. C. Edmunds, On certain finitely based varieties of semigroups, Semigroup Forum 15 (1977/78), 21-39.
    MathSciNet     CrossRef

  5. C. C. Edmunds, Varieties generated by semigroups of order four, Semigroup Forum 21 (1980), 67-81.
    MathSciNet     CrossRef

  6. C. F. Fennemore, All varieties of bands. I, II, Math. Nachr. 48 (1971), 237-252; ibid., 253-262.
    MathSciNet     CrossRef

  7. J. A. Gerhard, The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195-224.
    MathSciNet     CrossRef

  8. M. Jackson, Finite semigroups whose varieties have uncountably many subvarieties, J. Algebra 228 (2000), 512-535.
    MathSciNet     CrossRef

  9. E. W. H. Lee, Hereditarily finitely based monoids of extensive transformations, Algebra Universalis 61 (2009), 31-58.
    MathSciNet     CrossRef

  10. E. W. H. Lee, Varieties generated by 2-testable monoids, Studia Sci. Math. Hungar. 49 (2012), 366-389.
    MathSciNet     CrossRef

  11. E. W. H. Lee, Finite basis problem for semigroups of order five or less: generalization and revisitation, Studia Logica 101 (2013), 95-115.
    MathSciNet     CrossRef

  12. E. W. H. Lee and J. R. Li, Minimal non-finitely based monoids, Dissertationes Math. (Rozprawy Mat.) 475 (2011), 65 pp.
    MathSciNet     CrossRef

  13. E. W. H. Lee and W. T. Zhang, Finite basis problem for semigroups of order six, LMS J. Comput. Math. 18 (2015), 1-129
    MathSciNet     CrossRef

  14. J. R. Li and Y. F. Luo, Equational property of certain transformation monoids, Internat. J. Algebra Comput. 20 (2010), 833-845.
    MathSciNet     CrossRef

  15. J. R. Li, W. T. Zhang, and Y. F. Luo, On the finite basis problem for the variety generated by all n-element semigroups, Algebra Universalis 73 (2015), 225-248.
    MathSciNet     CrossRef

  16. Y. F. Luo and W. T. Zhang, On the variety generated by all semigroups of order three, J. Algebra 334 (2011), 1-30.
    MathSciNet     CrossRef

  17. S. Oates and M. B. Powell, Identical relations in finite groups, J. Algebra 1 (1964), 11-39.
    MathSciNet     CrossRef

  18. P. Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298-314.
    MathSciNet     CrossRef

  19. M. V. Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 319-340 (in Russian); English transl.: Math. USSR-Izv. 30 (1988), 295-314.
    MathSciNet    

  20. L. N. Shevrin and M. V. Volkov, Identities of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1985(11), 3-47 (in Russian); English transl.: Soviet Math. (Iz. VUZ) 29(11) (1985), 1-64.
    MathSciNet    

  21. The on-line encyclopedia of integer sequences, http://oeis.org/A058129

  22. A. N. Trahtman, Finiteness of a basis of identities of five-element semigroups, in: Semigroups and their homomorphisms (ed. E. S. Lyapin), Ross. Gos. Ped. Univ., Leningrad, 1991, 76-97 (in Russian).
    MathSciNet    

  23. M. V. Volkov, The finite basis property of varieties of semigroups, Mat. Zametki 45 (1989), 3, 12-23 (in Russian); English transl.: Math. Notes 45 (1989), 187-194.
    MathSciNet    

  24. M. V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn. 53 (2001), 171-199.
    MathSciNet    

  25. M. V. Volkov, Reflexive relations, extensive transformations and piecewise testable languages of a given height, Internat. J. Algebra Comput. 14 (2004), 817-827.
    MathSciNet    

Glasnik Matematicki Home Page