Nonfinitizability of classes of representable polyadic algebras

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.