Algebraic Logic Iv. Equality in Polyadic Algebras

Introduction. A standard way to begin the study of symbolic logic is to describe one after another the propositional calculus, the monadic functional calculus, the pure first-order functional calculus, and the functional calculus with equality. The algebraic aspects of these logical calculi belong to the theories of Boolean algebras, monadic algebras, polyadic algebras, and cylindric algebras respectively. The connection between the propositional calculus and Boolean algebras is well-known; for a recent exposition of it (and also of some aspects of the more advanced theories) see The basic concepts of algebraic logic, Amer. Math. Monthly vol. 63 (1956) pp. 363-387. Monadic algebras and polyadic algebras were studied in the first three papers of this sequence; [see Algebraic logic III, Trans. Amer. Math. Soc. vol. 83 (1956) pp. 430-470, and the references given there]. Cylindric algebras were introduced by Tarski and Thompson [Some general properties of cylindric algebras, Bull. Amer. Math. Soc. Abstract 58-1-85; see also Tarski, A representation theorem for cylindric algebras, Bull. Amer. Math. Soc. Abstract 58-1-86.] Most of what was done for polyadic algebras in [ll] and [ill] was restricted to locally finite polyadic algebras of infinite degree. (The Roman numerals refer to the other parts of this sequence.) Since it is known that every locally finite cylindric algebra of infinite degree possesses a natural polyadic structure, the results of those papers apply to cylindric algebras without any change. This paper, on the other hand, is mostly pre-cylindric; its main purpose is to discuss (in algebraic language) the introduction of equality. The paper is not self-contained. The notation introduced in §1 of [ill] will be used without any further explicit reference, and some of the basic concepts studied in [ill] (notably the concept of a predicate) will also be assumed known. (The most difficult part of [ill], the theory of terms, is used in §9 only.) At one point (§6) the representation theorem for simple polyadic algebras is needed, and later (§7) we make use of the duality theory for monadic algebras. Most parts of the paper, however (and, in particular, all definitions and the statements of all the theorems) are accessible to anyone who has skimmed through [III], provided that, in addition, he is acquainted with the elementary theory of functional polyadic algebras. For the con-