Algebra universalis

, 79:56

# Lee monoid $$L_4^1$$ is non-finitely based

• Inna A. Mikhailova
• Olga B. Sapir
Article

## 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 $$\ell \ge 3$$.

## Keywords

Lee monoids Identity Finite basis problem Non-finitely based Variety Isoterm

20M07 08B05

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© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Inna A. Mikhailova
• 1
• Olga B. Sapir
• 2
1. 1.Ural Federal UniversityEkaterinburgRussia
2. 2.NashvilleUSA

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