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.

scholarly journals Finitely generated pseudocomplemented distributive lattices

1975 ◽  
Vol 19 (2) ◽  
pp. 238-246 ◽  
Author(s):  
J. Berman ◽  
ph. Dwinger
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Distributive Lattices ◽  
Basic Fact ◽  
Finitely Generated ◽  
Partially Ordered ◽  
Finite Set ◽  
Natural Way

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

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.

scholarly journals The variety of semirings generated by distributive lattices and finite fields

2014 ◽  
Vol 95 (109) ◽  
pp. 101-109
Author(s):  
Yong Shao ◽  
Sinisa Crvenkovic ◽  
Melanija Mitrovic
Keyword(s):  
Finite Number ◽  
Distributive Lattice ◽  
Finite Fields ◽  
Distributive Lattices ◽  
Main Role ◽  
Subdirectly Irreducible ◽  
Finitely Based

A semiring variety is d-semisimple if it is generated by the distributive lattice of order two and a finite number of finite fields. A d-semisimple variety V = HSP{B2, F1,..., Fk} plays the main role in this paper. It will be proved that it is finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible members in it.

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.

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.

FREE TERNARY ALGEBRAS

2000 ◽  
Vol 10 (06) ◽  
pp. 739-749 ◽  
Author(s):  
RAYMOND BALBES
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Free Generator ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Bounded Distributive Lattice ◽  
Ternary Algebra ◽  
Free Generators ◽  
Ternary Algebras ◽  
Necessary And Sufficient

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

Varieties of involution monoids with extreme properties

2019 ◽  
Vol 70 (4) ◽  
pp. 1157-1180
Author(s):  
Edmond W H Lee
Keyword(s):  
The Other ◽  
Sufficient Condition ◽  
Finitely Based ◽  
Finitely Based Variety ◽  
Lattice Of Subvarieties ◽  
Power Set ◽  
Cross Variety ◽  
Positive Integers

Abstract A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These examples—all first of their kinds—include the following: finite involution monoids that generate huge varieties but whose reduct monoids generate Cross varieties; two finite involution monoids sharing a common reduct monoid such that one generates a huge, non-finitely based variety while the other generates a Cross variety; and two finite involution monoids that generate Cross varieties, the join of which is huge.

Subvarieties of the class of MS-algebras

1983 ◽  
Vol 95 (1-2) ◽  
pp. 157-169 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Distributive Lattice ◽  
Homomorphic Image ◽  
Lattice Of Subvarieties ◽  
De Morgan Algebras

SynopsisIn a previous publication (1983), we defined a class of algebras, denoted by MS, which generalises both de Morgan algebras and Stone algebras. Here we describe the lattice of subvarieties of MS. This is a 20-element distributive lattice. We then characterise all the subvarieties of MS by means of identities. We also show that some of these subvarieties can be described in terms of three important subsets of the algebra. Finally, we determine the greatest homomorphic image of an MS-algebra that belongs to a given subvariety.

scholarly journals Coproducts of Distributive Lattice based Algebras

10.29007/vx1v ◽  
2018 ◽  
Author(s):  
Leonardo Manuel Cabrer ◽  
Hilary Priestley
Keyword(s):  
Distributive Lattice ◽  
Systematic Study ◽  
Distributive Lattices ◽  
Finitely Generated ◽  
Cartesian Products

The analysis of coproducts in varieties of algebras has generally been variety-specific, relying on tools tailored to particular classes of algebras. A recurring theme, however, is the use of a categorical duality. Among the dualities and topological representations in the literature, natural dualities are particularly well behaved with respect to coproduct. Since (multisorted) natural dualities are based on hom-functors, they send coproducts into cartesian products.We carry out a systematic study of coproducts for finitely generated quasivarieties A that admit a (term) reduct in the variety D of bounded distributive lattices.

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