Paul R. Halmos. The basic concepts of algebraic logic. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 9–33. - Paul R. Halmos. Algebraic logic, I. Monadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 37–72. - Paul R. Halmos. Algebraic logic, II. Homogeneous locally finite polyadic Boolean algebras of infinite degree. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 97–166. - Paul R. Halmos. Algebraic logic, III. Predicates, terms, and operations in polyadic algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 169–209. - Paul R. Halmos. Algebraic logic, IV. Equality in polyadic algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 213–239. - Paul R. Halmos. Polyadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 243–248. - Paul R. Halmos. Predicates, terms, operations, and equality in polyadic Boolean algebras. Algebraic logic, by Paul R. Halmos, Chelsea Publishing Company, New York 1962, pp. 251–257.

1962 ◽  
Vol 27 (4) ◽  
pp. 469-470
Author(s):  
Aubert Daigneault
Keyword(s):  
New York ◽  
Algebraic Logic ◽  
Boolean Algebras ◽  
Publishing Company ◽  
Locally Finite ◽  
Polyadic Algebras ◽  
Infinite Degree ◽  
Basic Concepts

Copeland algebras

1972 ◽  
Vol 37 (4) ◽  
pp. 646-656 ◽  
Author(s):  
Daniel B. Demaree
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Boolean Algebras ◽  
Primary Concern ◽  
Cylindric Algebras ◽  
First Order ◽  
Polyadic Algebras ◽  
Algebraic Setting ◽  
Finite Set ◽  
Natural Way

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

Transformation Algebras

1961 ◽  
Vol 13 ◽  
pp. 602-613 ◽  
Author(s):  
Leon LeBlanc
Keyword(s):  
Elementary Theory ◽  
Locally Finite ◽  
Polyadic Algebras ◽  
Infinite Degree ◽  
Finite Transformation

The purpose of this paper is to show that most results concerning polyadic algebras can be generalized to transformation algebras. The results of this paper will clearly indicate that a great deal can be done in polyadic algebras without ever mentioning the quantifier structure (for instance, terms and operations can be characterized without the help of the quantifier structure, at least in the case where an equality is present). In § 1, we develop the elementary theory; in § 2, we study the different ways of extending a (locally finite) transformation algebra (of infinite degree) to a polyadic algebra; in § 3, we study equality transformation algebras; finally, in § 4, we show how terms and operations can be defined in equality transformation algebras.

Tensor products of polyadic algebras

1963 ◽  
Vol 28 (3) ◽  
pp. 177-200 ◽  
Author(s):  
Aubert Daigneault
Keyword(s):  
Boolean Algebra ◽  
Algebraic Logic ◽  
Algebraic Structures ◽  
Locally Finite ◽  
Elementary Equivalence ◽  
First Order ◽  
Polyadic Algebras ◽  
Logical Concept ◽  
Algebraic Treatment ◽  
Relational Systems

A basic concept of the theory of models is that of elementary equivalence of similar relational systems: two such systems are said to be elementarily equivalent if they satisfy the same first-order statements or, in other words, if they have the same (first-order) complete theory. It is possible to reformulate this notion of elementary equivalence of systems within the framework of algebraic logic by replacing theories by algebraic structures derived from them or more directly from the systems which are models of these theories. To any such theory T (or model of it), is indeed associated a locally finite polyadic algebra with equality, the underlying Boolean algebra of which is simply the well-known Tarski-Lindenbaum algebra of the theory. It is not hard to prove (see Section 6.1) that two systems are elementarily equivalent iff (i.e. if and only if) they have isomorphic polyadic. algebras. The possibility of replacing theories by algebraic structures and of reducing the purely logical concept of elementary equivalence to the algebraic one of isomorphism can be exploited to give a purely algebraic treatment of model-theoretic problems and suggests natural questions concerning these structures. The present paper illustrates that possibility.

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