Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T07:21:58.323Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW EXAMPLE OF A MINIMAL NONFINITELY BASED SEMIGROUP

Part of: Semigroups

Published online by Cambridge University Press:  06 September 2011

WEN TING ZHANG  and
YAN FENG LUO
WEN TING ZHANG*
Affiliation:
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China (email: [email protected])
YAN FENG LUO
Affiliation:
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Only three of the 15 973 distinct six-element semigroups are presently known to be nonfinitely based. This paper introduces a fourth example.

Keywords

semigroupsLee’s semigroupfinitely based

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 484 - 491
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This research was partially supported by the National Natural Science Foundation of China (No. 10971086) and the Fundamental Research Funds for the Central University (No. lzujbky-2009-119).

References

[1]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer, New York, 1981).CrossRefGoogle Scholar
[2]Distler, A. and Kelsey, T. W., ‘The monoids of orders eight, nine & ten’, Ann. Math. Artif. Intell. 56 (2009), 3–21.CrossRefGoogle Scholar
[3]Lee, E. W. H., ‘Finite basis problem for semigroups of order five or less: generalization and revisitation’, Studia Logica, to appear.Google Scholar
[4]Lee, E. W. H. and Li, J. R., ‘Minimal non-finitely based monoids’, Dissertationes Math. (Rozprawy Mat.) 475 (2011).Google Scholar
[5]Lee, E. W. H. and Volkov, M. V., ‘On the structure of the lattice of combinatorial Rees–Sushkevich varieties’, in: Semigroups and Formal Languages (eds. André, J. M.et al.) (World Scientific, Singapore, 2007), pp. 164–187.CrossRefGoogle Scholar
[6]Lee, E. W. H. and Volkov, M. V., ‘Limit varieties generated by completely 0-simple semigroups’, Internat. J. Algebra. Comput. 21 (2011), 257–294.CrossRefGoogle Scholar
[7]Mashevitskiĭ, G. I., ‘An example of a finite semigroup without an irreducible basis of identities in the class of completely 0-simple semigroups’, Uspekhi Mat. Nauk 38(2) (1983), 211–212 (in Russian); Russian Math. Surveys 38(2) (1983), 192–193 (English translation).Google Scholar
[8]Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969), 298–314.CrossRefGoogle Scholar
[9]Plemmons, R. J., ‘There are 15973 semigroups of order 6’, Math. Algorithms 2 (1967), 2–17.Google Scholar
[10]Sapir, M. V., ‘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); Math. USSR-Izv. 30 (1988), 295–314 (English translation).Google Scholar
[11]Shevrin, L. N. and Volkov, M. V., ‘Identities of semigroups’, Izv. Vyssh. Uchebn. Zaved. Mat. (11) (1985), 3–47 (in Russian); Soviet Math. (Iz. VUZ) 29(11) (1985), 1–64 (English translation).Google Scholar
[12]Trahtman, A. N., ‘Some finite infinitely basable semigroups’, Ural. Gos. Univ. Mat. Zap., Issl. Algebra. System, Sverdlovsk 14(2) (1987), 128–131 (in Russian).Google Scholar
[13]Trahtman, A. N., ‘Finiteness of identity bases of five-element semigroups’, in: Semigroups and their Homomorphisms (ed. Lyapin, E. S.) (Ross. Gos. Ped. Univ., Leningrad, 1991), pp. 76–97(in Russian).Google Scholar
[14]Volkov, M. V., ‘The finite basis question for varieties of semigroups’, Mat. Zametki 45(3) (1989), 12–23 (in Russian); Math. Notes 45(3) (1989), 187–194 (English translation).Google Scholar