Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T17:44:14.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite basis problem for semigroups of order six

Part of: Semigroups

Published online by Cambridge University Press:  01 January 2015

Edmond W. H. Lee  and
Wen Ting Zhang
Edmond W. H. Lee
Affiliation:
Division of Math, Science, and Technology, Nova Southeastern University, Fort Lauderdale, FL 33314, USA email edmond.lee@nova.edu
Wen Ting Zhang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu 730000, PR China email zhangwt@lzu.edu.cn

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.

Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist $15\,973$ pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15\,969$ distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.

MSC classification

Primary: 20M05: Free semigroups, generators and relations, word problems
Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 1 - 129
Copyright
© The Author(s) 2015 

References

Almeida, J., Finite semigroups and universal algebra (World Scientific, Singapore, 1994).Google Scholar
Bahturin, Yu. A. and Ol’šanskiĭ, A. Yu., ‘Identical relations in finite Lie rings’, Math. USSR-Sb. 25 (1975) 507–523 (Engl. transl. of Mat. Sb. 96 (1975) no. 138, 543–559).Google Scholar
Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambridge Philos. Soc. 31 (1935) 433–454.CrossRefGoogle Scholar
Bol’bot, A. D., ‘Finite basing of identities of four-element semigroups’, Sib. Math. J. 20 (1979) no. 2, 323.Google Scholar
Burris, S. and Sankappanavar, H. P., A course in universal algebra (Springer, New York, 1981).Google Scholar
Distler, A. and Kelsey, T. W., ‘The monoids of orders eight, nine & ten’, Ann. Math. Artif. Intell. 56 (2009) 3–25.Google Scholar
Distler, A. and Mitchell, J. D., ‘Smallsemi—a GAP package, version 0.6.2’, 2010, http://www.gap-system.org/Packages/smallsemi.html.Google Scholar
Edmunds, C. C., ‘Varieties generated by semigroups of order four’, Semigroup Forum 21 (1980) 67–81.Google Scholar
Edmunds, C. C., Lee, E. W. H. and Lee, K. W. K., ‘Small semigroups generating varieties with continuum many subvarieties’, Order 27 (2010) 83–100.Google Scholar
Golubov, È. A. and Sapir, M. V., ‘Varieties of finitely approximable semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 25–36 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1982) no. 11, 21–29).Google Scholar
Ježek, J., ‘Nonfinitely based three-element idempotent groupoids’, Algebra Universalis 20 (1985) 292–301.CrossRefGoogle Scholar
Karnofsky, J., ‘Finite equational bases for semigroups’, Notices Amer. Math. Soc. 17 (1970) 813–814.Google Scholar
Kruse, R. L., ‘Identities satisfied by a finite ring’, J. Algebra 26 (1973) 298–318.Google Scholar
Lee, E. W. H., ‘Identity bases for some non-exact varieties’, Semigroup Forum 68 (2004) 445–457.Google Scholar
Lee, E. W. H., ‘On identity bases of exclusion varieties for monoids’, Comm. Algebra 35 (2007) 2275–2280.CrossRefGoogle Scholar
Lee, E. W. H., ‘Combinatorial Rees–Sushkevich varieties are finitely based’, Internat. J. Algebra Comput. 18 (2008) 957–978.CrossRefGoogle Scholar
Lee, E. W. H., ‘Finite basis problem for 2-testable monoids’, Cent. Eur. J. Math. 9 (2011) 1–22.Google Scholar
Lee, E. W. H., ‘Finite basis problem for semigroups of order five or less: generalization and revisitation’, Studia Logica 101 (2013) 95–115.Google Scholar
Lee, E. W. H. and Li, J. R., ‘Minimal non-finitely based monoids’, Dissertationes Math. (Rozprawy Mat.) 475 (2011) 3–65.Google Scholar
Lee, E. W. H., Li, J. R. and Zhang, W. T., ‘Minimal non-finitely based semigroups’, Semigroup Forum 85 (2012) 577–580.Google Scholar
Lee, E. W. H. and Volkov, M. V., ‘On the structure of the lattice of combinatorial Rees–Sushkevich varieties’, Semigroups and formal languages, Proceedings of the International Conference, Lisboa, 2005 (eds André, J. M., Fernandes, V. H., Branco, M. J. J., Gomes, G. M. S., Fountain, J. and Meakin, J. C.; World Scientific, Singapore, 2007) 164–187.Google Scholar
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
Luo, Y. F. and Zhang, W. T., ‘On the variety generated by all semigroups of order three’, J. Algebra 334 (2011) 1–30.Google Scholar
L’vov, I. V., ‘Varieties of associative rings. I’, Algbra. Logic 12 (1973) 150–167 (Engl. transl. of Algebra i Logika 12 (1973) 269–297).Google Scholar
Lyndon, R. C., ‘Identities in two-valued calculi’, Trans. Amer. Math. Soc. 71 (1951) 457–465.Google Scholar
Lyndon, R. C., ‘Identities in finite algebras’, Proc. Amer. Math. Soc. 5 (1954) 8–9.Google Scholar
Mashevitskiĭ, G. I., ‘An example of a finite semigroup without an irreducible basis of identities in the class of completely 0-simple semigroups’, Russian Math. Surveys 38 (1983) no. 2, 192–193 (Engl. transl. of Uspekhi Mat. Nauk 38 (1983) no. 2, 211–212).Google Scholar
McKenzie, R., ‘Equational bases for lattice theories’, Math. Scand. 27 (1970) 24–38.Google Scholar
Murskiĭ, V. L., ‘The existence in the three-valued logic of a closed class with a finite basis, having no finite complete system of identities’, Soviet Math. Dokl. 6 (1965) 1020–1024 (Engl. transl. of Dokl. Akad. Nauk SSSR 163 (1965) 815–818).Google Scholar
Oates, S. and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964) 11–39.Google Scholar
Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969) 298–314.CrossRefGoogle Scholar
Petrich, M. and Reilly, N. R., Completely regular semigroups (Wiley & Sons, New York, 1999).Google Scholar
Plemmons, R. J., ‘There are 15973 semigroups of order 6’, Math. Alg. 2 (1967) 2–17.Google Scholar
Pollák, G., ‘On two classes of hereditarily finitely based semigroup identities’, Semigroup Forum 25 (1982) 9–33.Google Scholar
Pollák, G., ‘Some sufficient conditions for hereditarily finitely based varieties of semigroups’, Acta Sci. Math. (Szeged) 50 (1986) no. 3–4, 299–330.Google Scholar
Pollák, G. and Volkov, M. V., ‘On almost simple semigroup identities’, Colloq. Math. Soc. János Bolyai 39 (North-Holland, Amsterdam, 1985) 287–323.Google Scholar
Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 107–110 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1982) no. 11, 82–85).Google Scholar
Sapir, M. V., ‘Problems of Burnside type and the finite basis property in varieties of semigroups’, Math. USSR-Izv. 30 (1988) 295–314 (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 319–340).Google Scholar
Shevrin, L. N. and Volkov, M. V., ‘Identities of semigroups’, Soviet Math. (Iz. VUZ) 29 (1985) no. 11, 1–64 (Engl. transl. of Izv. Vyssh. Uchebn. Zaved. Mat. (1985) no. 11, 3–47).Google Scholar
Tarski, A., ‘Equational logic and equational theories of algebras’, Contributions to mathematical logic, Proceedings of the Logic Colloquium, Hannover, 1966 (eds Schmidt, H. A., Schütte, K. and Thiele, H. J.; North-Holland, Amsterdam, 1968) 275–288.Google Scholar
Tishchenko, A. V., ‘The finiteness of a base of identities for five-element monoids’, Semigroup Forum 20 (1980) 171–186.Google Scholar
Trahtman, A. N., ‘A basis of identities of the five-element Brandt semigroup’, Ural. Gos. Univ. Mat. Zap. 12 (1981) no. 3, 147–149 (in Russian).Google Scholar
Trahtman, A. N., ‘The finite basis question for semigroups of order less than six’, Semigroup Forum 27 (1983) 387–389.Google Scholar
Trahtman, A. N., ‘Some finite infinitely basable semigroups’, Ural. Gos. Univ. Mat. Zap. 14 (1987) no. 2, 128–131 (in Russian).Google Scholar
Trahtman, A. N., ‘Finiteness of identity bases of five-element semigroups’, Semigroups and their homomorphisms (ed. Lyapin, E. S.; Ross. Gos. Ped. Univ., Leningrad, 1991) 76–97 (in Russian).Google Scholar
Višin, V. V., ‘Identity transformations in a four-valued logic’, Soviet Math. Dokl. 4 (1963) 724–726 (Engl. transl. of Dokl. Akad. Nauk SSSR 150 (1963) 719–721).Google Scholar
Volkov, M. V., ‘The finite basis question for varieties of semigroups’, Math. Notes 45 (1989) no. 3, 187–194 (Engl. transl. of Mat. Zametki 45 (1989) no. 3, 12–23).Google Scholar
Volkov, M. V., ‘“Forbidden divisor” characterizations of epigroups with certain properties of group elements’, RIMS Kôkyûroku Bessatsu 1166 (2000) 226–234.Google Scholar
Volkov, M. V., ‘The finite basis problem for finite semigroups’, Sci. Math. Jpn. 53 (2001) 171–199.Google Scholar
Zhang, W. T. and Luo, Y. F., ‘A new example of a minimal non-finitely based semigroup’, Bull. Aust. Math. Soc. 84 (2011) 484–491.Google Scholar