Cylindric and polyadic algebras

1. Introduction. In recent years there have appeared two alge-braizations of the first-order predicate calculus; i.e., the polyadic algebras of Halmos [l; 2], and the cylindric algebras of Tarski [3; 4]. While polyadic algebras are the algebraic version of the pure first-order calculus, cylindric algebras yield an algebraization of the first-order calculus with equality. Since the pure calculus does not contain any identifiable predicate, one cannot expect to find the algebraic analogue of an equality predicate in a general polyadic algebra. It is reasonable, however, to consider "adjoining" an equality predicate, in some sense, to a polyadic algebra, and ask if one then obtains a cylindric algebra. This is the procedure followed here. An e-algebra is defined as a polyadic algebra with an equality predicate. We show that every e-algebra is in a natural way a cylindric algebra. Conversely , it is shown that in the presence of an infinite supply of variables and a local finiteness condition, cylindric algebras are in a natural way e-algebras, and the correspondence obtained in this way between e-algebras and cylindric algebras is one-to-one.