MS-almost distributive lattices

The purpose of this paper is to define and investigate a new equational class of algebras which we call MS-almost distributive lattices (MS-ADLs) as a common abstraction of De Morgan ADLs and Stone ADLs. It is observed that the class of MS-ADLs properly contains the class of MS-algebras, and most of the properties of MS-algebras are extended to the class of MS-ADLs.

Low Growth Equational Complexity

The equational complexity function $\beta \nu \,:\,{\open N} \to {\open N}$ of an equational class of algebras bounds the size of equation required to determine the membership of n-element algebras in . Known examples of finitely generated varieties with unbounded equational complexity have growth in Ω(nc), usually for c ≥ (1/2). We show that much slower growth is possible, exhibiting $O(\log_{2}^{3}(n))$ growth among varieties of semilattice-ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.

A Universal Variety of Point Algebras

Point algebras introduced by Evans are algebraic systems which capture the essence of multiplications (a,b) · (c,d)=(p,q) defined on the set of all ordered pairs of elements of a set S, where p and q are selected from among a,b,c,d by some well-defined rule. In 1961, Jonsson and Tarski gave an interesting example of a variety of algebras of type 〈2,1,1〉 for illustrating the failure of certain free algebra properties. In this paper, we show that this equational class of algebras, called the JT-variety, is a universal variety of point algebras in the sense that every variety generated by a point algebra is a reduct of the JT-variety.

Mapping Extension of an Algebra

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

Study of Convex Sublattices of a Lattice by a New Approach

The set of all convex sublattices CS(L) of a lattice L have been studied by a new approach. Introducing a new partial ordering relation "≤" it is shown that CS(L) is a lattice. Moreover L and CS(L) are in the same equational class. A number of properties of (CS(L); ≤) has also been included. Keywords: Convex sublattices; Standard element; Neutral element; Congruence.© 2009 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.DOI: 10.3329/jsr.v1i3.2484 J. Sci. Res. 1 (3), 558-562 (2009)

Equational spectrum of Hilbert varieties

We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.

DETERMINING WHETHER ${\mathsf V}({\bf A})$ HAS A MODEL COMPANION IS UNDECIDABLE

Using techniques pioneered by R. McKenzie, we prove that there is no algorithm which, given a finite algebra in a finite language, determines whether the variety (equational class) generated by the algebra has a model companion. In particular, there exists a finite algebra such that the variety it generates has no model companion; this answers a question of Burris and Werner from 1979.

Semi-de Morgan algebras

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Lakser's characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]).In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

Projective distributive p-algebras

A characterization of finite (weak) projectives in an equational class of distributive pseudocomplemented lattices is given. In the class of all such lattices, a finite lattice is projective if and only if its poset of join–irreducible elements forms a semilattice in which the minimal elements below the join of x and y are exactly the mininal elements below x or y. A similar condition works for any equational subclass.