Abstract
Since the 1970s, the smallest known non-finitely based involution semigroups are of order six. This paper exhibits the first example of a non-finitely based involution semigroup of order five.
Similar content being viewed by others
References
Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)
Auinger, K., Dolinka, I., Volkov, M.V.: Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. 14, 937–969 (2012)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)
Crvenković, S., Dolinka, I., Ésik, Z.: The variety of Kleene algebras with conversion is not finitely based. Theor. Comput. Sci. 230, 235–245 (2000)
Edmunds, C.C.: On certain finitely based varieties of semigroups. Semigroup Forum 15, 21–39 (1977)
Edmunds, C.C.: Varieties generated by semigroups of order four. Semigroup Forum 21, 67–81 (1980)
Gao, M., Zhang, W.T., Luo, Y.F.: The monoid of \(2 \times 2\) triangular boolean matrices under skew transposition is non-finitely based. Semigroup Forum 100, 153–168 (2020)
Jackson, M., Volkov, M.V.: The algebra of adjacency patterns: Rees matrix semigroups with reversion. In: Fields of Logic and Computation, Lecture Notes in Comput. Sci., 6300, pp. 414–443. Springer, Berlin (2010)
Jones, P.R.: The semigroups \(B_2\) and \(B_0\) are inherently nonfinitely based, as restriction semigroups. Int. J. Algebra Comput. 23, 1289–1335 (2013)
Kleĭman, E.I.: Bases of identities of varieties of inverse semigroups. Sibirsk. Mat. Zh. 20(4), 760–777 (1979). (translation in Siberian Math. J., 20, no. 4, 530–543 (1979))
Kruse, R.: Identities satisfied in a finite ring. J. Algebra 26, 298–318 (1973)
Lee, E.W.H.: On the variety generated by some monoid of order five. Acta Sci. Math. (Szeged) 74, 509–537 (2008)
Lee, E.W.H.: Finite basis problem for semigroups of order five or less: generalization and revisitation. Stud. Logica 101, 95–115 (2013)
Lee, E.W.H.: Finite involution semigroups with infinite irredundant bases of identities. Forum Math. 28, 587–607 (2016)
Lee, E.W.H.: Finitely based finite involution semigroups with non-finitely based reducts. Quaest. Math. 39, 217–243 (2016)
Lee, E.W.H.: Equational theories of unstable involution semigroups. Electron. Res. Announc. Math. Sci. 24, 10–20 (2017)
Lee, E.W.H.: A sufficient condition for the absence of irredundant bases. Houst. J. Math. 44, 399–411 (2018)
Lee, E.W.H.: Non-finitely based finite involution semigroups with finitely based semigroup reducts. Korean J. Math. 27, 53–62 (2019)
Lee, E.W.H., Li, J.R.: Minimal non-finitely based monoids. Diss. Math. 475, 65 (2011)
Lee, E.W.H., Li, J.R., Zhang, W.T.: Minimal non-finitely based semigroups. Semigroup Forum 85, 577–580 (2012)
Lee, E.W.H., Zhang, W.T.: Finite basis problem for semigroups of order six. LMS J. Comput. Math. 18, 1–129 (2015)
L’vov, I.V.: Varieties of assiciative rings I. Algebra i Logika 12, 269–297 (1973)
McKenzie, R.: Equational bases for lattice theories. Math. Scand. 27, 24–38 (1970)
Oates, S., Powell, M.B.: Identical relations in finite groups. J. Algebra 1, 11–39 (1964)
Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)
Tishchenko, A.V.: The finiteness of a base of identities for five-element monoids. Semigroup Forum 20, 171–186 (1980)
Trahtman, A.N.: Finiteness of identity bases of five-element semigroups. In: Lyapin, E.S. (ed.) Semigroups and Their Homomorphisms, pp. 76–97. Leningrad State Pedagogical Institute, Leningrad (1991) (Russian)
Zhang, W.T., Li, J.R., Luo, Y.F.: On the variety generated by the monoid of triangular \(2 \times 2\) matrices over a two-element field. Bull. Aust. Math. Soc. 86, 64–77 (2012)
Zhang, W.T., Luo, Y.F.: The finite basis problem for involution semigroups of triangular \( 2 \times 2\) matrices. Bull. Aust. Math. Soc. 101, 88–104 (2020)
Zhang, W.T., Luo, Y.F., Wang, N.: Finite basis problem for involution monoids of unitriangular Boolean matrices. Algebra Univers. 81, 7 (2020)
Acknowledgements
The authors would like to express their gratitude to Edmond W. H. Lee for his help in checking and revising this article, and to the reviewer for a number of helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by M. Jackson.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was partially supported by the National Natural Science Foundation of China under Grant nos. 11771191, 11401275, 11371177.
Rights and permissions
About this article
Cite this article
Gao, M., Zhang, W.T. & Luo, Y.F. A non-finitely based involution semigroup of order five. Algebra Univers. 81, 31 (2020). https://doi.org/10.1007/s00012-020-00662-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-020-00662-w