Cylindric Algebras of First - Order Languages

We show when two countable first-order languages have isomorphic cylindric algebras. Introduction. The cylindric algebra of a language is the co-dimensional cylindric algebra of formulas of the language modulo logical equivalence. We classify first-order languages according to the isomorphism types of their cylindric algebras. For two languages involving only predicate symbols, their cylindric algebras are isomorphic iff for every nEu, both languages have the same number of predicate symbols with at least n places. This solves the classification problem for free co-dimensional locally-finite cylindric algebras [4, Problem 2.8, p. 463] since these algebras are exactly the cylindric algebras of languages involving only predicate symbols. For any first-order theory, the topological space of its models with ECA classes (classes which consist of all models of some theory) as closed sets is a natural dual for the Lindenbaum-Tarski algebra of the theory. By enriching the class of models to the concrete category of models and isomorphisms and replacing the topology with the ultraproduct operations, we obtain a natural model-theoretic dual for the cylindric algebra of formulas of the theory. This duality is used in demonstrating that various properties of languages are invariant under cylindric isomorphism. The basic tool used in isomorphism construction is a decomposition version of the Cantor back-and-forth construction [7]. Notation. All languages, theories and sets of symbols will be countable. All theories will be first-order theories with equality. "Symbol" will mean nonlogical symbol. The type of an «-ary predicate symbol is n; that of an w-ary operation symbol is n + fa. Propositional letters are regarded as 0-ary predicate symbols and constants as 0-ary operation symbols. Let lAu = {XA, IVi, 2lA, . . .}. The {full) type sequence of a language is the sequence where a¡ is the number (possibly co) of symbols of the language of type i. The reduced type sequence is the sequence (b0, bx/i, bx, bx I/2, . . . > where b¡ is co if there are Received by the editors September 23, 1974. AMS (MOS) subject classifications (1970). Primary 02J15.