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COMBINATORIAL REES–SUSHKEVICH VARIETIES THAT ARE CROSS, FINITELY GENERATED, OR SMALL

Part of: Varieties Semigroups

Published online by Cambridge University Press:  13 August 2009

EDMOND W. H. LEE
EDMOND W. H. LEE*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada (email: ewl@sfu.ca)
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Abstract

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A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set ℱ of finitely generated varieties constitutes an incomplete sublattice and the set 𝒮 of small varieties constitutes a strict incomplete sublattice of ℱ. Consequently, a combinatorial Rees–Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set Σ of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity 𝒪(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ.

Keywords

semigroups0-simplevarietiesRees–SushkevichCrossfinitely basedfinitely generatedsmall

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups 08B15: Lattices of varieties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 64 - 84
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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