Abstract
Let n be a positive integer and \({\mathbf{S}_{n}}\) the variety generated by all semigroups of order n. It was shown by Volkov that \({\mathbf{S}_{n}}\) is non-finitely based for each \({n \geq 5}\) . It was shown by Luo and Zhang that \({\mathbf{S}_{3}}\) is finitely based. It is easy to show that the varieties \({\mathbf{S}_{1}, \mathbf{S}_{2}}\) are finitely based. However, the finite basis problem for \({\mathbf{S}_{4}}\) had been open for more than twenty years. In this paper, we solve this problem by showing that the variety \({\mathbf{S}_{4}}\) is finitely based. Moreover, we give an explicit finite basis for \({\mathbf{S}_{4}}\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bogdanović, S.: Semigroups with a system of subsemigroups. University of Novi Sad, Institute of Mathematics, Faculty of Science, Novi Sad (1985)
Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Springer, New York (1981)
Edmunds C.C.: Varieties generated by semigroups of order four. Semigroup Forum 21, 67–81 (1980)
Forsythe G.E.: SWAC computes 126 distinct semigroups of order 4. Proc. Amer. Math. Soc. 6, 443–447 (1955)
Howie, J.M.: Fundamentals of Semigroup Theory. Charendon Press, Oxford (1995)
Jackson, M.: Finite semigroups whose varieties have uncountably many subvarieties. J. Algebra 228, 512––535 (2000)
Lee, E.W.H.: Finite basis problem for semigroups of order five or less: generalization and revisitation. Studia Logica 101, 95–115 (2013)
Lee, E.W.H., Li, J.: Minimal non-finitely based monoids. Dissertationes Math. (Rozprawy Mat.) 475, 1–65 (2011)
Luo, Y., Zhang, W.: On the variety generated by all semigroups of order three. J. Algebra 334, 1–30 (2011)
Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)
Sapir, M.V.: Problems of Burnside type and the finite basis property in varieties of semigroups. Math. USSR Izv. 30(2), 295–314 (1988); translation of Izv. Akad. Nauk SSSR Ser. Mat. 51(2), 319–340 (1987)
Shevrin, L.N., Volkov, M.V.: Identities of semigroups. Izv. Vyssh. Uchebn. Zaved. Mat. 11, 3–47 (1985) (Russian)
Tamura, T.: Notes on finite semigroups and determination of semigroups of order 4. J. Gakugei. Tokushima Univ. Math. 5, 17–27 (1954)
Trahtman, A.N.: Finiteness of identity bases of five-element semigroups. In: Lyapin, E.S. (Ed.) Semigroups and their Homomorphisms, pp. 76–97. Ross. Gos. Ped. Univ., Leningrad (1991) (Russian)
Vernikov, B.M., Volkov, M.V.: Lattices of nilpotent varieties of semigroups, II. Izv. Ural. Gos. Univ. Mat. Mekh. 1(10), 13–33 (1998)
Volkov, M.V.: The finite basis question for varieties of semigroups. Math. Notes 45(3), 187–194 (1989), translation of Mat. Zametki 45(3), 12–23 (1989)
Volkov, M.V.: The finite basis problem for the finite semigroups. Sci. Math. Jpn. 53, 171–199 (2001)
Zhang, W., Luo, Y.: The variety generated by all non-permutative and non-idempotent semigroups of order four. In: Proceedings of the International Conference on Algebra 2010. Algebra, 721–735. World Sci. Publ., Hackensack, NJ (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Jackson
Presented by M. Jackson.
Dedicated to Professor Norman R. Reilly on the occasion of his 75th birthday
This research was partially supported by the NSF of China (No. 10971086, 11371177, 11401274) and by the NSF of Gansu Province (No. 1107RJZA218).
Rights and permissions
About this article
Cite this article
Li, J.R., Zhang, W.T. & Luo, Y.F. On the finite basis problem for the variety generated by all n-element semigroups. Algebra Univers. 73, 225–248 (2015). https://doi.org/10.1007/s00012-015-0326-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-015-0326-3