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

Reilly, Norman R.

doi:10.1007/s00012-008-2091-z

Cited by 14 publications

(15 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Part (i) follows from Graham [1, Corollary 2], and part (ii) follows from Reilly [10, Theorem 7.2(xiv)]. 2 [9] Varieties generated by completely 0-simple semigroups 383…”

Section: Construction Of a Covermentioning

confidence: 99%

“…(i) This solution to the word problem for B 2 can be found in Reilly [9]. It can also be derived from the somewhat different solution provided by Mashevitzky [6].…”

Section: Introductionmentioning

confidence: 99%

“…We adopt the notation: LZ = V (L 2 ), RZ = V (R 2 ), A [3] Varieties generated by completely 0-simple semigroups 377 [11] was the first to present a basis of identities for B 2 . His proof contains a small lacuna, which can be fixed, however, in a number of ways, one of which can be found in [9,Theorem 7.4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Varieties Generated by Completely 0-Simple Semigroups

Reilly¹

2008

JAZ

Self Cite

View full text Add to dashboard Cite

Kublanovsky has shown that if a subvariety V of the variety RS n generated by completely 0-simple semigroups over groups of exponent n is itself generated by completely 0-simple semigroups, then it must satisfy one of three conditions:The conditions (i) and (ii) are also sufficient conditions. In this note, we complete Kublanovsky's programme by refining condition (iii) to obtain a complete set of conditions that are both necessary and sufficient.2000 Mathematics subject classification: 20M07, 08B15.

show abstract

“…Part (i) follows from Graham [1, Corollary 2], and part (ii) follows from Reilly [10, Theorem 7.2(xiv)]. 2 [9] Varieties generated by completely 0-simple semigroups 383…”

Section: Construction Of a Covermentioning

confidence: 99%

“…(i) This solution to the word problem for B 2 can be found in Reilly [9]. It can also be derived from the somewhat different solution provided by Mashevitzky [6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Varieties Generated by Completely 0-Simple Semigroups

Reilly¹

2008

JAZ

Self Cite

View full text Add to dashboard Cite

show abstract

“…The basis was first found by Trahtman [32]; according to Reilly [27], there was a small lacuna in the proof, which the latter closed in the cited paper. Not only is B 2 finitely based, it is hereditarily finitely based, that is, every subvariety of the variety it generates is finitely based [20,Corollary 3.8].…”

mentioning

confidence: 92%

THE SEMIGROUPS B₂AND B₀ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

Jones

2013

Int. J. Algebra Comput.

View full text Add to dashboard Cite

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

show abstract

“…The study of combinatorial Rees-Sushkevich varieties is thus precisely the study of subvarieties of A 2 . These varieties have recently been investigated by Reilly, Volkov, and the author (see, for example, [10,11,14,21,28]). Unlike the general case in which many Rees-Sushkevich varieties containing nontrivial groups are nonfinitely based, all subvarieties of A 2 are finitely based [11].…”

Section: Introductionmentioning

confidence: 99%

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

Lee

2009

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 14 publications

References 7 publications

Varieties Generated by Completely 0-Simple Semigroups

Varieties Generated by Completely 0-Simple Semigroups

THE SEMIGROUPS B₂AND B₀ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

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

Contact Info

Product

Resources

About

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

Cited by 14 publications

References 7 publications

Varieties Generated by Completely 0-Simple Semigroups

Varieties Generated by Completely 0-Simple Semigroups

THE SEMIGROUPS B2AND B0ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

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

Contact Info

Product

Resources

About

THE SEMIGROUPS B₂AND B₀ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS