Complete Congruences on the Lattice of Rees–Sushkevich Varieties

Reilly, Norman R.

doi:10.1080/00927870701451375

Cited by 10 publications

(3 citation statements)

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“…In [9], the second author built on this pioneering work by developing a framework on the lattice of Rees-Sushkevich varieties in the form of a complete congruence.…”

Section: Introductionmentioning

confidence: 99%

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Centrality in Rees-Sushkevich varieties

Lee

Reilly

2008

Algebra univers.

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Denote by RSn the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RSn 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 CmRSn to be the class of all semigroups S in RSn 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 CmRSn constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C 2 RSn 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 CmRSn is also addressed. For any fixed n ≥ 2, it is shown that although the varieties CmRSn, where m = 1, 2, . . . , are all finitely based, their complete intersection (denoted by C∞RSn) is non-finitely based. Further, the variety C∞RSn contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based.

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“…In [9], the second author built on this pioneering work by developing a framework on the lattice of Rees-Sushkevich varieties in the form of a complete congruence.…”

Section: Introductionmentioning

confidence: 99%

“…Notation and background material are presented in Section 2. A complete congruence Θ that is crucial in the study of the lattice LE(RS n ) of exact Rees-Sushkevich varieties (in [9,10]) is recalled in this section. In Section 3, the varieties C m RS n , where m = 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Centrality in Rees-Sushkevich varieties

Lee

Reilly

2008

Algebra univers.

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“…Investigation of the lattice of Rees-Sushkevich varieties has recently been initiated by Reilly, Volkov, and the author (see [5]- [10], [12]- [14], and [19]). In particular, several aspects of the lattice C of combinatorial Rees-Sushkevich varieties have been considered in [5]- [7], [10], and [19].…”

Section: Introductionmentioning

confidence: 99%

On the complete join of permutative combinatorial Rees-Sushkevich varieties

Lee

2007

ija

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A semigroup variety is a Rees-Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. The collection of all permutative combinatorial Rees-Sushkevich varieties constitutes an incomplete lattice that does not contain the complete join J of all its varieties. The objective of this article is to investigate the subvarieties of J. It is shown that J is locally finite, non-finitely generated, and contains only finitely based subvarieties. The subvarieties of J are precisely the combinatorial Rees-Sushkevich varieties that do not contain a certain semigroup of order four.Mathematics Subject Classification: 20M07, 08B15.

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Historical Overview and Main Results

Lee¹

2022

Frontiers in Mathematics

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Complete Congruences on the Lattice of Rees–Sushkevich Varieties

Cited by 10 publications

References 12 publications

Centrality in Rees-Sushkevich varieties

Centrality in Rees-Sushkevich varieties

On the complete join of permutative combinatorial Rees-Sushkevich varieties

Historical Overview and Main Results

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