Varieties Generated by Completely 0-Simple Semigroups

Reilly, Norman R.

doi:10.1017/s1446788708000311

Cited by 15 publications

(10 citation statements)

References 8 publications

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“…It is also possible to characterize the exact subvarieties of RS n in terms of intervals in L(RS n ). 12], Corollary 9.2). Let P n denote the set of positive prime divisors of n. Then the exact subvarieties of RS n are precisely the varieties in the following intervals:…”

Section: Introductionmentioning

confidence: 94%

“…An important feature of Rees-Sushkevich varieties is that some are exact (that is, generated by completely simple or completely 0-simple semigroups) while others are not. The second important step was taken when Kublanovsky [4] and the second author [12] characterized the exact varieties in terms of the inclusion or exclusion of certain finite semigroups.…”

Section: Introductionmentioning

confidence: 99%

“…It is clear that the join α∈A U α is again an exact variety. But the intersection α∈A U α need not be an exact variety (see [12,Section 7]). However, if we let V be denote the variety generated by all the completely 0-simple semigroups in α∈A U α , then V is exact and is the largest exact variety contained in α∈A U α .…”

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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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Centrality in Rees-Sushkevich varieties

Lee

Reilly

2008

Algebra univers.

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“…We introduce a new type of multiple clustering systems, or clusterers, based on Rees matrix semigroups, which are well-known technical tools of semigroup theory (see [18]). Let us also refer, for example, to [21,22,34] for recent results concerning this construction. The class of all Brandt semigroups is a proper subclass of the class of all Rees matrix semigroups.…”

Section: Introductionmentioning

confidence: 99%

Rees Matrix Constructions for Clustering of Data

Kelarev¹,

Watters²,

Yearwood³

2009

J. Aust. Math. Soc.

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This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.2000 Mathematics subject classification: primary 16S36, 20M35; secondary 20M25, 68T.

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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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Varieties Generated by Completely 0-Simple Semigroups

Cited by 15 publications

References 8 publications

Centrality in Rees-Sushkevich varieties

Centrality in Rees-Sushkevich varieties

Rees Matrix Constructions for Clustering of Data

On the complete join of permutative combinatorial Rees-Sushkevich varieties

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