Willem J. Blok

Equivalence of Consequence Operations

This paper is based on Lectures 1, 2 and 4 in the series of ten lectures titled “Algebraic Structures for Logic” that Professor Blok and I presented at the Twenty Third Holiday Mathematics Symposium held at New Mexico State University in Las Cruces, New Mexico, January 8-12, 1999. These three lectures presented a new approach to the algebraization of deductive systems, and after the symposium we made plans to publish a joint paper, to be written by Blok, further developing these ideas. That project was still incomplete when Blok died. In fact, there is no indication that he had prepared a draft of the paper, and we do not know what new material he intended to include. I am therefore not in a position to complete the project as he had envisioned it. So, I have settled for the more limited objective of presenting the material from the three lectures, leaving to others the task of adapting the techniques used there to more general situations.

ASSERTIONALLY EQUIVALENT QUASIVARIETIES

A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation τ naturally induces a deductive system, called the τ-assertional logic of K. Two quasivarieties are τ-assertionally equivalent if they have the same τ-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as τ-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation.

Free łukasiewicz and hoop residuation algebras

Hoop residuation algebras are the {→, 1}-subreducts of hoops; they include Hilbert algebras and the {→, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which is a copy of the {→, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops).

A finite basis theorem for quasivarieties

We consider varieties with the property that the intersection of any pair of principal congruences is finitely generated, and, in fact, generated by pairs of terms constructed from the generators of the principal components in a uniform way. We say that varieties with this property haveequationally definable principal meets (EDPM). There are many examples of these varieties occurring in the literature, especially in connection with metalogical investigations. The main result of this paper is that every finite, subdirectly irreducible member of a variety with EDPM generates a finitely based quasivariety. This is proved in Section 2. In the first section we prove that every variety with EDPM is congruence-distributive.

Fragments of r-Mingle

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociński, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations.

The Fraser-Horn and Apple properties

We consider varieties f in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with f a family of derived varieties, d( f ): a variety in d( f ) is generated by algebras A where the universe of A consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra B E f and the operations of A are those of B that preserve this congruence class. We also consider the prime variety of <, denoted %0, generated by all finite simple algebras in f. We show how the structure of finite algebras in f is determined to a considerable extent by %0 and d( f ). In particular, the free Walgebra on n generators, F<(n), has as many directly indecomposable factors as F<0(n) and the structure of these factors is determined by the varieties d( f ). This allows us to produce in many cases explicit formulas for the cardinality of F<(n). Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme. The literature on free algebras in varieties of algebras is quite extensive. Most of the work done in this area, however, concerns the structure or cardinality of the free algebras in specific varieties, while only a few attempts have been made to develop a more general theory which might explain the structure of free algebras in certain classes of varieties in more general terms. For example, Quackenbush [1974] did this for discriminator varieties, and Plonka for regular varieties [1971]. Recently, Cornish [1983], using a categorical approach, described the free algebras in varieties whose algebras admit an involution operator. In the present paper we study a class of locally finite varieties whose finite free algebras are, to a considerable degree, determined by the finite free algebras in simpler varieties, naturally associated with the original ones. The class we have in mind consists of the varieties which have the Fraser-Horn Property (FHP for short) and what we call the Apple Property (AP for short). A variety has the Fraser-Horn Property if there are no skew congruences on any direct product of a finite number of algebras in the variety. If the congruence lattice of an algebra A has a unique coatom i.e., is isomorphic to L fE3 1, where L is a lattice with 0,1 then clearly A is directly indecomposable. A variety f has the Apple Property if the converse holds as well for all finite algebras; that is, if the finite Received by the editors October 7,1985. 1980 Mathematics Subject Classification. Primary 08A05, 08B20, 08A40; Secondary 03G25, 06D20.

On the lattice of quasivarieties of Sugihara algebras

AbstractLet S denote the variety of Sugihara algebras. We prove that the lattice Λ (K) of subquasivarieties of a given quasivariety K

Algebras Defined from Ordered Sets and the Varieties they Generate

\subseteq