Lee monoid \(L_4^1\) is non-finitely based
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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 \(\ell \ge 3\).
KeywordsLee monoids Identity Finite basis problem Non-finitely based Variety Isoterm
Mathematics Subject Classification20M07 08B05
The authors thank an anonymous referee for helpful comments.
- 4.Lee, E.W.H.: A sufficient condition for the absence of irredundant bases. Houston J. Math. 44(2), 399–411 (2018)Google Scholar