Bjarni Jónsson

On the canonicity of Sahlqvist identities

We give a simple proof of the canonicity of Sahlqvist identities, using methods that were introduced in a paper by Jónsson and Tarski in 1951.

Sublattices of a free lattice

Professor R. A. Dean has proved (1, Theorem 3) that a completely free lattice generated by a countable partially ordered set is isomorphic to a sublattice of a free lattice. In particular, it follows that a free product of countably many countable chains can be isomorphically embedded in a free lattice. Generalizing this we show (2.1) that the class of all lattices that can be isomorphically embedded in free lattices is closed under the operation of forming free lattice-products with arbitrarily many factors. We also prove (2.4) that this class is closed under the operation of forming simply ordered sums with denumerably many summands. Finally we show (2.7) that every finite dimensional sublattice of a free lattice is finite.

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.

EXTENSIONS OF RELATIONAL STRUCTURES

Publisher Summary This chapter discusses extensions of relational structures. Model theory has abundantly demonstrated that such investigations are worthwhile and can lead to interesting results. It is only natural that one should there be primarily concerned with elementary classes, or at least with L-classes for some language L, and that the axioms should be in some sense of a model-theoretic character. However, certain properties of a different nature. It is frequently necessary to exclude from the discussion certain trivial classes of structures. It is known that the amalgamation property holds for each of the following classes : groups, groups with a given operator domain, abelian groups, fields, differential fields of characteristic 0, partially ordered sets, lattices, Boolean algebras, locally finite-dimensional cylindric algebras of a given infinite dimension. On the other hand, the property is known to fail for the class of all semi-groups and for the class of all rings. A critical point in any attempt at such a generalization is, of course, the choice of definitions of the basic concepts, particularly of the notion of an algebraic extension.

A REPORT ON SUBLATTICES OF A FREE LATTICE

Publisher Summary This chapter presents a report on sublattices of a free lattice. It elaborates the characterization problem for finite sublattices of a free lattice. It is conjectured that these are precisely the finite lattices that satisfy three first order conditions, the Whitman condition. Every finite sublattice of a free lattice is an S-lattice, for all three conditions hold in free lattices, and is the key condition in Whitmans famous characterization of free lattices, and is easy consequences of Whitmans results. It is an open question whether, conversely, every S -lattice is embeddable in a free lattice. Several characterizations of finite sublattices of free lattices have been found and some of these even characterize finitely generated sublattices of free lattices, but none of them is in terms of a finite set of first-order formulas. A lattice L is transferable if for every embedding f of L into the ideal lattice of a lattice K , there exists and embedding g of L into K.

A Survey of Boolean Algebras with Operators

The purpose of this survey is to call attention to the unifying role of the concept of a Boolean algebra with operators. The first chapter contains a brief history of this concept and a list of theorems from universal algebra that play an important role in the treatment of the subject. The next two chapters are concerned with the general theory, in particular canonical extensions and dualities, while each of the remaining chapters treats a particular class of BAO’s. The presentation is highly incomplete. In each chapter some basic concepts are introduced and their properties are illustrated by stating a number of key theorems, mostly without proofs. The bibliography contains only a miniscule portion of the relevant literature, but should be sufficient to open up the subject to a reader who wants to pursue a particular topic further.

Free algebras in discriminator varieties

We investigate V-free algebras onn generators,Fn=Fr(V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindricalgebras. Sample questions are: (a) IsFn+1 embeddable inFn? (b) DoesFn contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.

Relatively free proucts in regular varieties

The principal result states that ifV is a regular variety without operations of rank zero (distinguished elements), then any two representations of an algebraA ∈V as aV-free product of subalgebras ofA have a common refinement.

Arithmetic of Ordered Sets

The fundamental operations of ordinal and cardinal arithmetic have been extended to arbitrary binary relations and their isomorphism types. (Whitehead and Russell [1912], pp. 291 ff.) When applied to (partially) ordered sets, these operations yield new ordered sets. A unified treatment of these operations was initiated in Birkhoff [1937] and [1942], where it was shown that most of the basic laws of arithmetic apply in this general setting. See also Birkhoff [1967], pp. 55 ff.

Relation algebras and Schro¨der categories

Abstract We describe a construction that embeds a finite system of simple relation algebras A,i, i ∈ I , in a simple relation algebra A, not as subalgebras of A, but as relative subalgebras. Schroder categories will play a useful role in this construction. As an application, it is shown that every relation algebra that is generated by a single equivalence element is finite and representable.