## Mathematics Faculty Articles

#### Title

Centrality in Rees–Sushkevich Varieties

#### Document Type

Article

#### Publication Date

3-2008

#### Publication Title

Algebra Universalis

#### Keywords

Semigroups, Complete o-simple, Varieties, Central, Rees-Sushkevich

#### ISSN

0002-5240

#### Volume

58

#### Issue/No.

2

#### First Page

145

#### Last Page

180

#### Abstract

Denote by **RS**_{n} the variety generated by all completely 0-simple semigroups over groups of exponent dividing *n*. Subvarieties of **RS**_{n} are called Rees–Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define **C**_{m}**RS**_{n} to be the class of all semigroups *S* in **RS**_{n} with the property that if the product of *m* idempotents of *S* belongs to some subgroup of *S*, then the product belongs to the center of that subgroup. The classes **C**_{m}**RS**_{n} constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of **C**_{2}**RS**_{n} is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing *n*. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties **C**_{m}**RS**_{n} is also addressed. For any fixed *n* ≥ 2, it is shown that although the varieties **C**_{m}**RS**_{n}, where *m* = 1, 2, ... , are all finitely based, their complete intersection (denoted by **C**_{∞}**RS**_{n}) is non-finitely based. Further, the variety **C**_{∞}**RS**_{n} contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based.

#### NSUWorks Citation

Lee, Edmond W. H. and Reilly, Norman R., "Centrality in Rees–Sushkevich Varieties" (2008). *Mathematics Faculty Articles*. 157.

https://nsuworks.nova.edu/math_facarticles/157

#### ORCID ID

0000-0002-1662-3734

#### ResearcherID

I-6970-2013

#### DOI

10.1007/s00012-008-2054-4

## Comments

© Birkhaueser 2008