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FINITE BASIS PROBLEM FOR INVOLUTION MONOIDS OF ORDER FIVE

Part of: Semigroups

Published online by Cambridge University Press:  09 October 2023

BIN BIN HAN Open the ORCID record for BIN BIN HAN [Opens in a new window] ,
WEN TING ZHANG Open the ORCID record for WEN TING ZHANG [Opens in a new window]  and
YAN FENG LUO Open the ORCID record for YAN FENG LUO [Opens in a new window]
BIN BIN HAN
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: hanbb19@lzu.edu.cn
WEN TING ZHANG*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China
YAN FENG LUO
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: luoyf@lzu.edu.cn
*
e-mail: zhangwt@lzu.edu.cn
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Abstract

An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.

Keywords

monoidinvolutionidentity basisfinite basis problem

MSC classification

Primary: 20M05: Free semigroups, generators and relations, word problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 350 - 364
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The authors are partially supported by the National Natural Science Foundation of China (Nos. 12271224, 12171213, 12161062) and the Fundamental Research Funds for the Central University (No. lzujbky-2023-ey06).

References

Adair, C., ‘Bands with an involution’, J. Algebra 75 (1982), 297–314.CrossRefGoogle Scholar
Bahturin, Y. A. and Ol’šanskiĭ, A. Y., ‘Identical relations in finite Lie rings’, Math. USSR-Sb. 25 (1975), 507–523; translation of Mat. Sb. (N.S.) 96(138) (1975), 543–559.CrossRefGoogle Scholar
Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer, New York, 1981).CrossRefGoogle Scholar
Edmunds, C. C., ‘On certain finitely based varieties of semigroups’, Semigroup Forum 15 (1977), 21–39.CrossRefGoogle Scholar
Gao, M., Zhang, W. T. and Luo, Y. F., ‘A non-finitely based involution semigroup of order five’, Algebra Universalis 81 (2020), Paper no. 31.CrossRefGoogle Scholar
Jackson, M. and Volkov, M. V., ‘The algebra of adjacency patterns: Rees matrix semigroups with reversion’, in: Fields of Logic and Computation (eds. Blass, A., Dershowitz, N. and Reisig, W.) (Springer, Berlin, 2010), 414–443.CrossRefGoogle Scholar
Kruse, R., ‘Identities satisfied in a finite ring’, J. Algebra 26 (1973), 298–318.CrossRefGoogle Scholar
L’vov, I. V., ‘Varieties of assiciative rings I’, Algebra i Logika 12 (1973), 269–297.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.CrossRefGoogle Scholar
Lee, E. W. H., ‘Finitely based finite involution semigroups with nonfinitely based reducts’, Quaest. Math. 39 (2016), 217–243.CrossRefGoogle Scholar
Lee, E. W. H., ‘Equational theories of unstable involution semigroups’, Electron. Res. Announc. Math. Sci. 24 (2017), 10–20.Google Scholar
Lee, E. W. H., ‘Non-finitely based finite involution semigroups with finitely based semigroup reducts’, Korean J. Math. 27 (2019), 53–62.Google Scholar
Lee, E. W. H., ‘Varieties of involution monoids with extreme properties’, Q. J. Math. 70 (2019), 1157–1180.CrossRefGoogle Scholar
Lee, E. W. H., ‘Intervals of varieties of involution semigroups with contrasting reduct intervals’, Boll. Unione Mat. Ital. 15 (2022), 527–540.CrossRefGoogle Scholar
Lee, E. W. H., Advances in the Theory of Varieties of Semigroups (Birkhäuser, Cham, 2023).CrossRefGoogle Scholar
Lee, E. W. H. and Li, J. R., ‘Minimal non-finitely based monoids’, Dissertationes Math. 475 (2011), 65 pages.CrossRefGoogle Scholar
Lee, E. W. H., Li, J. R. and Zhang, W. T., ‘Minimal non-finitely based semigroups’, Semigroup Forum 85 (2012), 577–580.CrossRefGoogle Scholar
Lee, E. W. H. and Zhang, W. T., ‘Finite basis problem for semigroups of order six’, LMS J. Comput. Math. 18(1) (2015), 1–129.CrossRefGoogle Scholar
McKenzie, R., ‘Equational bases for lattice theories’, Math. Scand. 27 (1970), 24–38.CrossRefGoogle Scholar
Oates, S. and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964), 11–39.CrossRefGoogle Scholar
Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969), 298–314.CrossRefGoogle Scholar
Trahtman, A. N., ‘Finiteness of identity bases of five-element semigroups’, in: Semigroups and Their Homomorphisms (ed. Lyapin, E. S.) (Leningrad State Pedagogical Institute, Leningrad, 1991), 76–97 (in Russian).Google Scholar