Mathematics Faculty Articles

Combinatorial Rees–Sushkevich Varieties That Are Cross, Finitely Generated, Or Small

Document Type

Article

Publication Date

2-2010

Publication Title

Bulletin of the Australian Mathematical Society

Keywords

Primary 20M07, Secondary 03C05, 08B15

ISSN

0004-9727

Volume

81

Issue/No.

1

First Page

64

Last Page

84

Abstract

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 � 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 �(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ.

ORCID ID

0000-0002-1662-3734

ResearcherID

I-6970-2013

DOI

10.1017/S0004972709000616

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