Some conditions related to the exactness of Rees-Sushkevich varieties

Kublanovsky, Stanislav I.; Lee, Edmond W. H.; Reilly, Norman R.

doi:10.1007/s00233-007-9003-y

Cited by 17 publications

(10 citation statements)

References 6 publications

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“…Some indication of this can be found in Petrich [7], Sapir [9] and Shevrin and Volkov [10]. Both the varieties B 2 and NB 2 are very important in the characterization of those varieties that are generated by completely 0-simple semigroups; see Lee and Reilly [4] and Reilly [8]. The idea behind much of the work in this note is contained in the following simple observation.…”

Section: Vol 59 2008mentioning

confidence: 99%

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

Reilly

2008

Algebra univers.

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We provide new solutions to the word problems for the variety B 2 generated by the five element Brandt semigroup B 2 with zero divisors and the variety NB 2 generated by B 2 and the two element left and right zero semigroups. We also provide a finite basis of identities for the variety NB 2 . This leads to a complete description of the interval [B 2 , NB 2 ], in the lattice of semigroup varieties. This is a critical interval in the study of the lattice of aperiodic Rees-Sushkevich varieties. In addition, the varieties B 2 and NB 2 are especially important in the characterization of those Rees-Sushkevich varieties that are exact, that is, generated by completely 0-simple semigroups. Byproducts of this note are techniques that lend themselves to a deeper study of the structure of the free objects in the varieties in the interval [B 2 , NB 2 ].

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Section: Vol 59 2008mentioning

confidence: 99%

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

Reilly

2008

Algebra univers.

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“…Following Kublanovsky et al (2008) we call any element of RS n a ReesSushkevich variety. We say that a semigroup variety is exact (respectively, non-exact) if it is a Rees-Sushkevich variety generated by (respectively, not generated by) completely (0-) simple semigroups.…”

Section: Introductionmentioning

confidence: 99%

Complete Congruences on the Lattice of Rees–Sushkevich Varieties

Reilly

2007

Communications in Algebra

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We study the lattice RS n of subvarieties of the variety of semigroups generated by completely 0-simple semigroups over groups with exponent dividing n, with a particular focus on the lattice RS n consisting of those varieties that are generated by completely 0-simple semigroups. The sublattice of RS n consisting of the aperiodic varieties is described and several endomorphisms of RS n considered. The complete congruence on RS n that relates varieties containing the same aperiodic completely 0-simple semigroups is considered in some detail.

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“…The class of 0-simple semigroups was one of the first classes of semigroups to be studied, in the pioneering work of Rees [20] and Sushkevich [23], and remains one of the most important and interesting classes of semigroups. Following Kublanovsky [6,7], any subvariety of a periodic variety generated by 0-simple semigroups is referred to as a Rees-Sushkevich variety. One of the most important results concerning Rees-Sushkevich varieties, due to Mashevitzky [18] and Hall et al [5], is that for each integer n ≥ 1, the variety generated by all 0-simple semigroups over groups of exponent dividing n is finitely based.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Rees–sushkevich Varieties That Are Cross, Finitely Generated, or Small

Lee

2009

Bull. Aust. Math. Soc.

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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 S of small varieties constitutes a strict incomplete sublattice of F. Consequently, a combinatorial ReesSushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set of identities defines, within the largest combinatorial Rees-Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity O(nk) where n is the number of identities in and k is the length of the longest word in .2000 Mathematics subject classification: primary 20M07; secondary 03C05, 08B15.

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Some conditions related to the exactness of Rees-Sushkevich varieties

Cited by 17 publications

References 6 publications

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

Complete Congruences on the Lattice of Rees–Sushkevich Varieties

Combinatorial Rees–sushkevich Varieties That Are Cross, Finitely Generated, or Small

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