Skip to main content Accessibility help
×
Home

Finite basis problem for semigroups of order six

Part of: Semigroups

Published online by Cambridge University Press:  01 January 2015

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
Corresponding

Abstract

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.

Type
Research Article
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) 507523 (Engl. transl. of Mat. Sb. 96 (1975) no. 138, 543–559).CrossRefGoogle Scholar
Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambridge Philos. Soc. 31 (1935) 433454.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).CrossRefGoogle Scholar
Distler, A. and Kelsey, T. W., ‘The monoids of orders eight, nine & ten’, Ann. Math. Artif. Intell. 56 (2009) 325.CrossRefGoogle 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) 6781.CrossRefGoogle Scholar
Edmunds, C. C., Lee, E. W. H. and Lee, K. W. K., ‘Small semigroups generating varieties with continuum many subvarieties’, Order 27 (2010) 83100.CrossRefGoogle Scholar
Golubov, È. A. and Sapir, M. V., ‘Varieties of finitely approximable semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 2536 (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) 292301.CrossRefGoogle Scholar
Karnofsky, J., ‘Finite equational bases for semigroups’, Notices Amer. Math. Soc. 17 (1970) 813814.Google Scholar
Kruse, R. L., ‘Identities satisfied by a finite ring’, J. Algebra 26 (1973) 298318.CrossRefGoogle Scholar
Lee, E. W. H., ‘Identity bases for some non-exact varieties’, Semigroup Forum 68 (2004) 445457.CrossRefGoogle Scholar
Lee, E. W. H., ‘On identity bases of exclusion varieties for monoids’, Comm. Algebra 35 (2007) 22752280.CrossRefGoogle Scholar
Lee, E. W. H., ‘Combinatorial Rees–Sushkevich varieties are finitely based’, Internat. J. Algebra Comput. 18 (2008) 957978.CrossRefGoogle Scholar
Lee, E. W. H., ‘Finite basis problem for 2-testable monoids’, Cent. Eur. J. Math. 9 (2011) 122.CrossRefGoogle Scholar
Lee, E. W. H., ‘Finite basis problem for semigroups of order five or less: generalization and revisitation’, Studia Logica 101 (2013) 95115.CrossRefGoogle Scholar
Lee, E. W. H. and Li, J. R., ‘Minimal non-finitely based monoids’, Dissertationes Math. (Rozprawy Mat.) 475 (2011) 365.Google Scholar
Lee, E. W. H., Li, J. R. and Zhang, W. T., ‘Minimal non-finitely based semigroups’, Semigroup Forum 85 (2012) 577580.CrossRefGoogle 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) 164187.Google Scholar
Lee, E. W. H. and Volkov, M. V., ‘Limit varieties generated by completely 0-simple semigroups’, Internat. J. Algebra. Comput. 21 (2011) 257294.CrossRefGoogle Scholar
Luo, Y. F. and Zhang, W. T., ‘On the variety generated by all semigroups of order three’, J. Algebra 334 (2011) 130.CrossRefGoogle Scholar
L’vov, I. V., ‘Varieties of associative rings. I’, Algbra. Logic 12 (1973) 150167 (Engl. transl. of Algebra i Logika 12 (1973) 269–297).CrossRefGoogle Scholar
Lyndon, R. C., ‘Identities in two-valued calculi’, Trans. Amer. Math. Soc. 71 (1951) 457465.CrossRefGoogle Scholar
Lyndon, R. C., ‘Identities in finite algebras’, Proc. Amer. Math. Soc. 5 (1954) 89.CrossRefGoogle 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, 192193 (Engl. transl. of Uspekhi Mat. Nauk 38 (1983) no. 2, 211–212).CrossRefGoogle Scholar
McKenzie, R., ‘Equational bases for lattice theories’, Math. Scand. 27 (1970) 2438.CrossRefGoogle 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) 10201024 (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) 1139.CrossRefGoogle Scholar
Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969) 298314.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) 217.Google Scholar
Pollák, G., ‘On two classes of hereditarily finitely based semigroup identities’, Semigroup Forum 25 (1982) 933.CrossRefGoogle Scholar
Pollák, G., ‘Some sufficient conditions for hereditarily finitely based varieties of semigroups’, Acta Sci. Math. (Szeged) 50 (1986) no. 3–4, 299330.Google Scholar
Pollák, G. and Volkov, M. V., ‘On almost simple semigroup identities’, Colloq. Math. Soc. János Bolyai 39 (North-Holland, Amsterdam, 1985) 287323.Google Scholar
Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Soviet Math. (Iz. VUZ) 26 (1982) no. 11, 107110 (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) 295314 (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 319–340).CrossRefGoogle Scholar
Shevrin, L. N. and Volkov, M. V., ‘Identities of semigroups’, Soviet Math. (Iz. VUZ) 29 (1985) no. 11, 164 (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) 275288.CrossRefGoogle Scholar
Tishchenko, A. V., ‘The finiteness of a base of identities for five-element monoids’, Semigroup Forum 20 (1980) 171186.CrossRefGoogle Scholar
Trahtman, A. N., ‘A basis of identities of the five-element Brandt semigroup’, Ural. Gos. Univ. Mat. Zap. 12 (1981) no. 3, 147149 (in Russian).Google Scholar
Trahtman, A. N., ‘The finite basis question for semigroups of order less than six’, Semigroup Forum 27 (1983) 387389.CrossRefGoogle Scholar
Trahtman, A. N., ‘Some finite infinitely basable semigroups’, Ural. Gos. Univ. Mat. Zap. 14 (1987) no. 2, 128131 (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) 7697 (in Russian).Google Scholar
Višin, V. V., ‘Identity transformations in a four-valued logic’, Soviet Math. Dokl. 4 (1963) 724726 (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, 187194 (Engl. transl. of Mat. Zametki 45 (1989) no. 3, 12–23).CrossRefGoogle Scholar
Volkov, M. V., ‘“Forbidden divisor” characterizations of epigroups with certain properties of group elements’, RIMS Kôkyûroku Bessatsu 1166 (2000) 226234.Google Scholar
Volkov, M. V., ‘The finite basis problem for finite semigroups’, Sci. Math. Jpn. 53 (2001) 171199.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) 484491.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 207 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 27th January 2021. This data will be updated every 24 hours.

Access
Hostname: page-component-898fc554b-t4g97 Total loading time: 0.456 Render date: 2021-01-27T02:05:03.500Z Query parameters: { "hasAccess": "1", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Finite basis problem for semigroups of order six
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Finite basis problem for semigroups of order six
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Finite basis problem for semigroups of order six
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *