FINITELY GENERATED LIMIT VARIETIES OF APERIODIC MONOIDS WITH CENTRAL IDEMPOTENTS

2009 ◽  
Vol 08 (06) ◽  
pp. 779-796 ◽  
Author(s):  
EDMOND W. H. LEE
Keyword(s):  
Present Article ◽  
Variety Of Algebras ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finitely Based Variety ◽  
Limit Variety ◽  
Aperiodic Monoids

A non-finitely based variety of algebras is said to be a limit variety if all its proper subvarieties are finitely based. Recently, Marcel Jackson published two examples of finitely generated limit varieties of aperiodic monoids with central idempotents and questioned whether or not they are unique. The present article answers this question affirmatively.

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

A Variety Where the Set of Subalgebras of Finite Simple Algebras is Not Recursive

1998 ◽  
Vol 08 (06) ◽  
pp. 681-688 ◽  
Author(s):  
Stanislav Kublanovsky ◽  
Mark Sapir
Keyword(s):  
Variety Of Algebras ◽  
Finitely Based ◽  
Binary Operations ◽  
Finitely Based Variety ◽  
Simple Algebras

We construct a finitely based variety of algebras with two binary operations where the set of subalgebras of finite simple algebras is not recursive.

scholarly journals Limit varieties of aperiodic monoids with commuting idempotents

2020 ◽  
pp. 2150160
Author(s):  
S. V. Gusev
Keyword(s):  
Variety Of Algebras ◽  
Finitely Based ◽  
Aperiodic Monoids ◽  
Nonfinitely Based

A variety of algebras is called limit if it is nonfinitely-based but all its proper subvarieties are finitely-based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.

A JUST NON-FINITELY BASED VARIETY OF BIGROUPS

2001 ◽  
Vol 29 (9) ◽  
pp. 4011-4046 ◽  
Author(s):  
C. K. Gupta* ◽  
A. N. Krasilnikov
Keyword(s):  
Finitely Based ◽  
Finitely Based Variety

Levi and Malcev Theorems for Finite-Dimensional Algebras from the Variety Defined by the Identities x2 = J( x,y, zu) =0

2021 ◽  
Vol 28 (01) ◽  
pp. 87-90
Author(s):  
Óscar Guajardo Garza ◽  
Marina Rasskazova ◽  
Liudmila Sabinina
Keyword(s):  
Lie Algebras ◽  
Present Article ◽  
Variety Of Algebras ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

We study the variety of binary Lie algebras defined by the identities [Formula: see text], where [Formula: see text] denotes the Jacobian of [Formula: see text], [Formula: see text], [Formula: see text]. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

scholarly journals On some varieties of ai-semirings satisfying xp+1 ≈ x

2018 ◽  
Vol 16 (1) ◽  
pp. 913-923
Author(s):  
Aifa Wang ◽  
Yong Shao
Keyword(s):  
Distributive Lattice ◽  
Finitely Based ◽  
Finitely Generated ◽  
Lattice Of Subvarieties

AbstractThe aim of this paper is to study the lattice of subvarieties of the ai-semiring variety defined by the additional identities$$\begin{array}{} \displaystyle x^{p+1}\approx x\,\,\mbox{and}\,\,zxyz\approx(zxzyz)^{p}zyxz(zxzyz)^{p}, \end{array} $$wherepis a prime. It is shown that this lattice is a distributive lattice of order 179. Also, each member of this lattice is finitely based and finitely generated.

A non-finitely-based variety of center-by-metabelian pointed groups

2014 ◽  
Vol 95 (5-6) ◽  
pp. 743-746
Author(s):  
G. S. Deryabina ◽  
A. N. Krasil’nikov
Keyword(s):  
Finitely Based ◽  
Finitely Based Variety

scholarly journals On the Schur-Baer property

1981 ◽  
Vol 31 (3) ◽  
pp. 343-361 ◽  
Author(s):  
Mohammad Reza R. Moghaddam
Keyword(s):  
Factor Group ◽  
Finitely Based ◽  
Verbal Subgroup ◽  
Finitely Based Variety ◽  
Precise Bound

AbstractIn 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

scholarly journals On product varieties of groups

1971 ◽  
Vol 5 (2) ◽  
pp. 239-240 ◽  
Author(s):  
M.R. Vaughan-Lee
Keyword(s):  
Finitely Based ◽  
Finitely Based Variety ◽  
Variety Of Groups

An example is given of a finitely based variety of groups such that is not finitely based.Let be the variety of groups determined by the laws (1) [[x1, x2], [x3, x4, [x5, x6]], (2) [[x1, x2, x3], [x4, x5]] [[x1x2], [x4, x5, x3]]−1, [[x1, x2, x3], [x1, x2]]. Then is not finitely based.

scholarly journals Some finitely based varieties of rings

1974 ◽  
Vol 17 (2) ◽  
pp. 246-255 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Nilpotent Group ◽  
Finitely Based ◽  
Finitely Generated ◽  
Nilpotent Variety ◽  
Moufang Loops ◽  
Associative Rings ◽  
Finitely Related

The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.

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