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 \({\mathbf{S}_{4}}\) .
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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).
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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
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DOI: https://doi.org/10.1007/s00012-015-0326-3