Discriminator Varieties Research Articles

A finitely axiomatizable undecidable equational theory with recursively solvable word problems

In this paper we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty (1992). The construction produces a discriminator variety with the aforementioned properties starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature. This result also presents a common generalization of the earlier results obtained by B. Wells (1982) and A. Mekler, E. Nelson, and S. Shelah (1993).

Universal classes of simple relation algebras

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the formU×Ufor some non-empty setU—is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-calledgroup relation algebras) is also universal.We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes calledLyndon algebras) and those constructed from modular lattices with a zero (sometimes calledMaddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6.

Rich and poor algebras

We prove that there exist finitely generated, simple, rich algebras in discriminator varieties having continuum many finitely generated mutually nonembeddable algebras, under some additional assumptions imposed on the varieties.

Birkhoff-like sheaf representation for varieties of lattice expansions

Given a variety ν we study the existence of a class ℱ such that S1 every A e ν can be represented as a global subdirect product with factors in ℱ and S2 every non-trivial A e ℱ is globally indecomposable. We show that the following varieties (and its subvarieties) have a class ℱ satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.

Skew Boolean algebras and discriminator varieties

We investigate the class of skew Boolean algebras which are also meet semilattices under the natural skew lattice partial order. Such algebras, called hereskew Boolean ∩-algebras, are quite common. Indeed, any algebra A in a discriminator variety with a constant term has a skew Boolean ∩-algebra polynomial reduct whose congruences coincide with those of A.

Free algebras in discriminator varieties

In this paper we prove a representation theorem for the free algebras of certain discriminator varieties which arise when analyzing Boolean products. This result applies to several varieties such as relatively complemented distributive lattices, (generalized) Boolean algebras, Post algebras, P-algebras, B-algebras and in general every variety of R5 lattices. In Section 1 we introduce notation and terminology. In Section 2 we extend the definition of JCl-spectral variety given in [10] to certain V3 classes and we give a characterization of the quasivarieties of a discriminator variety. In Section 3 we prove a theorem which represents the free algebras of the rig-spectral variety as certain Boolean products. In Section 4 we give some applications of this representation theorem.

Locally Boolean spectra

In [35] we study three types of representability of algebras by sheaves: the global representability, the Stone representability and the locally Boolean representability. In that paper the study is centered in the two first of the three notions cited above, and the characterizations are given in terms of properties of the congruence lattice of the algebra. This paper is a continuation of [35] in which we study with the same approach the third of the above three notions: the locally Boolean representability. The central concepts of [35] are those of compactness of a set of congruences, projectives and (appropriate versions of) the Chinese Remainder Theorem. An unexpected fact is that the first two of the three concepts above are precisely the central concepts in the study of the locally Boolean representability. The generality of the results obtained in both works puts in perspective the power of the universal algebra in the study of sheaf representations of algebras. As it can be noted in the applications, in all cases the results are obtained by combining a minute amount of particular algebra with strong general universal algebra theorems. In Section 1 we fix notation and terminology and review a few basic facts which will be needed later. In Section 2 we define the notion of compactness of a set of congruences and prove a key result of the paper (Lemma 2.1). In Section 3 we define locally Boolean subdirect products and show the relation between this concept and sheaves. In Section 4 we develop, in a similar way as in Section 4 of [35], the general construction presented in [26], and we obtain some results on compactness of a set of congruences. In Section 5 we associate with each universal class de' a discriminator variety called the Jg-spectral variety which is such that the locally Boolean representations with /'actors in X-{trivial algebras} are just the irredundant fully expanded subdirect representations by simple algebras in the tit-spectral variety (Theorem 5.3). As an application we obtain that if ~ ' is a V3 class, then the class of locally Boolean d/Z-representable algebras is closed under ultraproducts and direct products. Finally, we use a result of R. McKenzie to

Decidable discriminator varieties with lattice stalks

We determine those universal classes of lattices which generate a decidable discriminator variety when augmented by a ternary discriminator term. They are the locally finite universal classes whose finite members are almost homogeneous.

On the structure of varieties with equationally definable principal congruences IV

The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) ≡ z (modΘ(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z ∈ A} is a transversal of the set of equivalence classes of Θ(a, b); (3)p(a, b, z) andp(a′,b′,z) define the same transversal wheneverΘ(a,b)=Θ(a′,b′); (4)Θ(p(x, y, 1), 1)= Θ(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.

Embeddability skeletons of non-locally finite discriminator varieties with a poor algebra

Embeddability skeletons of non-locally finite discriminator varieties with a poor algebra

Compact algebras in discriminator varieties

Compact algebras in discriminator varieties

Decidable Discriminator Varieties from Unary Classes

Let $\mathcal {K}$ be a class of (universal) algebras of fixed type. ${\mathcal {K}^t}$ denotes the class obtained by augmenting each member of $\mathcal {K}$ by the ternary discriminator function $(t(x,y,z) = x$ if $x \ne y,t(x,x,z) = z)$, while $\vee ({\mathcal {K}^t})$ is the closure of ${\mathcal {K}^t}$ under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to $\vee ({\mathcal {K}^t})$ where $\mathcal {K}$ consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some $\vee ({\mathcal {K}^t})$ is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes $\mathcal {K}$ of unary algebras of finite type for which the first-order theory of $\vee ({\mathcal {K}^t})$ is decidable.

Discriminator varieties of Boolean algebras with residuated operators

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral ralgebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jonsson. 1. Preliminaries. A unary operation f on a Boolean algebra A0 = (A,+, 0, ·, 1,− ) is additive if f(x + y) = f(x) + f(y) and normal if f(0) = 0. For an n-ary operation f on A0, a sequence a ∈ A and i < n we define the (a, i)-translate of f to be the unary operation fa,i(x) = f(a0, . . . , ai−1, x, ai+1, . . . , an−1) . An operator on A0 is an n-ary operation for which all (a, i)-translates are additive and normal. Note that 0-ary operations (constants) have no translates, so they are operators by default. A = (A0,F) is a Boolean algebra with operators (BAO for short) if each f ∈ F is an operator on A0. The arity (or rank) of f is denoted by %f . To be an operator on a Boolean algebra is of course an equational property, and the variety of all BAOs with operators in F will be denoted by BAOF . The variety BAO{f}, where f is a unary operator, is usually referred to as the variety of modal algebras (the algebraic counterpart of modal logic). 1991 Mathematics Subject Classification: Primary 06E25; Secondary 03G15, 08A40, 03B45. The paper is in final form and no version of it will be published elsewhere.

Decidable discriminator varieties from unary classes

Let K \mathcal {K} be a class of (universal) algebras of fixed type. K t {\mathcal {K}^t} denotes the class obtained by augmenting each member of K \mathcal {K} by the ternary discriminator function ( t ( x , y , z ) = x (t(x,y,z) = x if x ≠ y , t ( x , x , z ) = z ) x \ne y,t(x,x,z) = z) , while ∨ ( K t ) \vee ({\mathcal {K}^t}) is the closure of K t {\mathcal {K}^t} under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to ∨ ( K t ) \vee ({\mathcal {K}^t}) where K \mathcal {K} consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some ∨ ( K t ) \vee ({\mathcal {K}^t}) is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes K \mathcal {K} of unary algebras of finite type for which the first-order theory of ∨ ( K t ) \vee ({\mathcal {K}^t}) is decidable.

Countable skeletons of finitely generated discriminator varieties

Countable skeletons of finitely generated discriminator varieties

Discriminator varieties and symbolic computation

We look at two aspects of discriminator varieties which could be of considerable interest in symbolic computation: 1. discriminator varieties are unitary (i.e., there is always a most general unifier of two unifiable terms), and 2. every mathematical problem can be routinely cast in the form † p 1 ≈ q 1, …, p k ≈ q k implies the equation x ≈ y. Item (l) offers possibilities for implementations in computational logic, and (2) shows that Birkhoff's five rules of inference for equational logic are all one needs to prove theorems in mathematics.

Elementary theory of embeddability skeletons of discriminator varieties

Elementary theory of embeddability skeletons of discriminator varieties

The word problem for discriminator varieties

The word problem for discriminator varieties

Decidable discriminator varieties from unary varieties

AbstractWe determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

Decompositions of dual discriminator varieties

A necessary and sufficient condition for a dual discriminator variety to be the product of a discriminator variety and a dual discriminator variety is given, along with an example of such a variety.