Algebra universalis

, 79:56 | Cite as

Lee monoid \(L_4^1\) is non-finitely based

  • Inna A. Mikhailova
  • Olga B. Sapir


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 \(\ell \ge 3\).


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

Mathematics Subject Classification

20M07 08B05 



The authors thank an anonymous referee for helpful comments.


  1. 1.
    Edmunds, C.C.: On certain finitely based varieties of semigroups. Semigroup Forum 15(1), 21–39 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Edmunds, C.C.: Varieties generated by semigroups of order four. Semigroup Forum 21(1), 67–81 (1980)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jackson, M.G.: Finiteness properties of varieties and the restriction to finite algebras. Semigroup Forum 70, 159–187 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lee, E.W.H.: A sufficient condition for the absence of irredundant bases. Houston J. Math. 44(2), 399–411 (2018)Google Scholar
  5. 5.
    Lee, E.W.H.: On a class of completely join prime J-trivial semigroups with unique involution. Algebra Universalis 78, 131–145 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    McKenzie, R.N.: Tarski’s finite basis problem is undecidable. Internat. J. Algebra Comput. 6, 49–104 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Perkins, P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sapir, O.B.: Non-finitely based monoids. Semigroup Forum 90(3), 557–586 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sapir, O.B.: Lee monoids are non-finitely based while the sets of their isoterms are finitely based. Bull. Aust. Math. Soc. 97(3), 422–434 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Volkov, M.V.: The finite basis problem for finite semigroups. Sci. Math. Jpn. 53, 171–199 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zhang, W.T., Luo, Y.F.: A new example of non-finitely based semigroups. Bull. Aust. Math. Soc. 84, 484–491 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, W.T.: Existence of a new limit variety of aperiodic monoids. Semigroup Forum 86, 212–220 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations