Polyadic algebra

Abstract: The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras. In the immediately preceding paper (6), Donald Monk proves that for 3 < a, < co the class RCAa: is not finitely axiomatizable. In ?3 we extend this result to the classes RPAo, RPEAo, and RDfa:. In ?1 we study some relationships between these classes which are essential to the later work, but have some independent interest. In ?2 some examples of nonrepresentable algebras with special properties are given. Sections 1 and 2 should be comprehendible by anyone with a knowledge of algebraic logic. In ?3 we assume an intimate knowledge of the immediately pre- ceding paper of Monk. In ?4 we state some problems. I am grateful to Professor Donald Monk for his help and encouragement and for making available at an early date the results on which this work was based.

Abstract: Publisher Summary This chapter discusses the representation theorems for logical probabilities. One of the main theorems determines each probability c on the language L in terms of probabilities related to models of L . The theorem roughly states that any probability c (α) is obtained by taking a “linear combination” of probabilities in models with respect to some “weight-function” λ on the set of all models. Another main theorem gives a simple version of de Finetti's theorem on exchangeable events as stated within the framework of logical probabilities. De Finetti's theorem on exchangeable events roughly states that the probability of any such event is obtained by taking a “linear combination” of probabilities corresponding to cases of independent equiprobable events. The representation theorem for logical probabilities may be considered as a generalization of this result. The chapter provides a brief description of concepts and results from the theory of polyadic algebras needed for the subsequent development. The introduction of the theory of polyadic algebras is provided in the chapter. A polyadic algebra is the algebraic counterpart of the first order predicate logic obtained by identifying equivalent formulas. It consists of a Boolean algebra, nonempty set, and maps.

Abstract: The main result of the paper is that for I an infinite set, the class of polyadic I-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic I-algebras have a given common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra. Polyadic algebras were introduced by Halmos to provide an algebraic reflection of the study of first order logic without equality; later the algebras were enriched to allow the discussion of equality. That the notion is an adequate reflection of first order logic was demonstrated by Halmos' representation theorem for locally finite polyadic algebras of infinite degree (with or without equality). Daigneault and Monk have proved a strong extension of Halmos' theorem; namely, every polyadic algebra of infinite degree (without equality) is representable. Thus the notion of polyadic algebra is an adequate reflection of Keisler's predicate logic having infinitary predictates. It is an interesting question to ask for algebraic versions of various model theoretic results. Daigneault has been successful in stating and proving algebraic versions of Beth's and Craig's theorems. This was done by proving the algebraic analogue of Robinson's consistency theorem: Localy finite polyadic I-algebras (with equality) of infinite degree have the amalgamation property. The major result of the present work is to remove the locally finite condition from Daigneault's result. With the stronger result, Robinson's, Beth's, and Craig's theorems follow for Keisler's logic though we shall defer this to a later paper. We shall preface our work with an outline of the basic theory of polyadic algebras including theorems of Halmos [10] and important dilation and compression results of Daigneault and Monk [5]. Our set theoretic notation is standard, but it is perhaps worthwhile to outline some of our conventions. If X and Y are two sets, we write YX for the set of all functions from Y into X. We shall often identify 2X with Xx X-the cartesian Presented to the Society, April 8, 1967; received by the editors February 27, 1969. AMS Subject Classifications. Primary 0240, 0248; Secondary 0235, 0250.