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On the rank of the Rees-Sushkevich varieties


Author: S. I. Kublanovskiĭ
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 23 (2011), nomer 4.
Journal: St. Petersburg Math. J. 23 (2012), 679-730
MSC (2010): Primary 20M07
DOI: https://doi.org/10.1090/S1061-0022-2012-01214-8
Published electronically: April 13, 2012
MathSciNet review: 2893522
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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.


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Additional Information

S. I. Kublanovskiĭ
Affiliation: TPO Severny Ochag, B. Konyushennaya 15, Office 30, St. Petersburg 191186, Russia
Email: stas1107@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01214-8
Keywords: Semigroups, $0$-prime, identity, rank, Rees–Sushkevich variety, Cross semigroup, finitely based semigroup, finite generation, smallness, polynomial algorithm
Received by editor(s): December 13, 2009
Published electronically: April 13, 2012
Article copyright: © Copyright 2012 American Mathematical Society