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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Relatively inherently nonfinitely q-based semigroups

Authors: Marcel Jackson and Mikhail Volkov
Journal: Trans. Amer. Math. Soc. 361 (2009), 2181-2206
MSC (2000): Primary 08C15, 20M20
Published electronically: November 25, 2008
MathSciNet review: 2465833
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Abstract: We prove that every semigroup $ {\bf S}$ whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds:

  • $ {\bf S}$ is finite;
  • $ {\bf S}$ has a faithful representation by injective partial maps on a set;
  • $ {\bf S}$ has a faithful representation by order preserving maps on a chain.
As a corollary it is shown that, in an asymptotic sense, almost all finite semigroups and finite monoids admit no finite basis for their quasi-identities.

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Additional Information

Marcel Jackson
Affiliation: Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

Mikhail Volkov
Affiliation: Department of Mathematics, Ural State University, Ekaterinburg 620083, Russia

Keywords: Quasi-identity, quasivariety, universal class, semigroup, injective map, order preserving map, finite q-basis property, inherently nonfinitely q-based semigroup relative to a class, 3-nilpotent semigroup, homotopy embedding
Received by editor(s): June 4, 2007
Published electronically: November 25, 2008
Additional Notes: The first author was supported by ARC Discovery Project Grant DP0342459
The second author acknowledges support from the Russian Foundation for Basic Research, grants 05-01-00540 and 06-01-00613. The paper was initiated during the second author’s Distinguished Fellowship at the Institute for Advanced Study of La Trobe University.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.