Abstract
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid \(L_4^1\) is non-finitely based. The monoid \(L_4^1\) was the only unsolved case in the finite basis problem for Lee monoids \(L_\ell ^1\), obtained by adjoining an identity element to the semigroup \(L_\ell \) generated by two idempotents a and b subjected to the relation \(0=abab \cdots \) (length \(\ell \)). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new proof to the results of Zhang–Luo and Lee that the semigroup \(L_\ell \) is non-finitely based for each \(\ell \ge 3\).
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Acknowledgements
The authors thank an anonymous referee for helpful comments.
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The research of the first author was supported by the Russian Foundation for Basic Research, project no. 17-01-00551, the Ministry of Education and Science of the Russian Federation, project no. 1.3253.2017, and the Competitiveness Program of Ural Federal University.
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Mikhailova, I.A., Sapir, O.B. Lee monoid \(L_4^1\) is non-finitely based. Algebra Univers. 79, 56 (2018). https://doi.org/10.1007/s00012-018-0541-9
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DOI: https://doi.org/10.1007/s00012-018-0541-9