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UNCOUNTABLY MANY VARIETIES OF COMPLETELY SIMPLE SEMIGROUPS WITH METABELIAN SUBGROUPS

Published online by Cambridge University Press:  06 December 2006

JIŘÍ KAĎOUREK
JIŘÍ KAĎOUREK
Affiliation:
Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, 602 00 Brno, Czech Republic e-mail: kadourek@math.muni.cz
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Abstract

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It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety $\mathcal{V}$ of completely simple semigroups the class of all idempotent generated members of $\mathcal{V}$ is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.

Keywords

Primary 20M07Secondary 20E10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 48 , Issue 3 , September 2006 , pp. 365 - 410
Copyright
2006 Glasgow Mathematical Journal Trust