On the rank of the Rees–Sushkevich varieties
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S. I. Kublanovskiĭ
Translated by: B. M. Bekker - St. Petersburg Math. J. 23 (2012), 679-730
- DOI: https://doi.org/10.1090/S1061-0022-2012-01214-8
- Published electronically: April 13, 2012
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Abstract:
A specific numerical characteristic of a variety of semigroups, the rank, is introduced. It is proved that the Rees–Sushkevich varieties with the same derivative, i.e., containing the same $0$-simple semigroups, are determined by their rank uniquely up to permutation identities. As a consequence, answers to several well-known questions are obtained. In particular, a description is given for the Rees–Sushkevich varieties satisfying finiteness conditions (finiteness of the base of identities or of the lattice of subvarieties, generation by a finite semigroup or by a completely $0$-simple semigroup, the condition of maximality, minimality, finite width, etc.). Some applications of an algorithmic nature are presented. In particular, it is shown that a Rees–Sushkevich variety defined by a finite set of identities or by a finite semigroup has a decidable (polynomially decidable) equational theory if and only if its derivative has the same property. This holds true for combinatorial varieties.References
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Bibliographic Information
- S. I. Kublanovskiĭ
- Affiliation: TPO Severny Ochag, B. Konyushennaya 15, Office 30, St. Petersburg 191186, Russia
- Email: stas1107@mail.ru
- Received by editor(s): December 13, 2009
- Published electronically: April 13, 2012
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 679-730
- MSC (2010): Primary 20M07
- DOI: https://doi.org/10.1090/S1061-0022-2012-01214-8
- MathSciNet review: 2893522