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Levi’s Commutator Theorems for Cancellative Semigroups

Abstract

For every semigroup of finite exponent whose chains of idempotents are uniformly bounded we construct an identity which holds on this semigroup but does not hold on the variety of all idempotent semigroups. This shows that the variety of all idempotent semigroups E is not contained in any finitely generated variety of semigroups. Since E is locally finite and each proper subvariety of E is finitely generated [1, 3, 4], the variety of all idempotent semigroups is a minimal example of an inherently non-finitely generated variety.

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Correspondence to Olga Sapir.

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Sapir, O. Levi’s Commutator Theorems for Cancellative Semigroups. Semigroup Forum 71, 140–146 (2005). https://doi.org/10.1007/s00233-004-0178-1

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  • DOI: https://doi.org/10.1007/s00233-004-0178-1

Keywords

  • Cancellative Semigroup
  • Proper Subvariety
  • Idempotent Semigroup
  • Commutator Theorem
  • Finite Exponent