Epimorphisms in cylindric algebras and definability in finite variable logic

Abstract

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of “Cylindric Set algebras”, by Henkin, et al. for finite n. By a result of Nemeti, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is greater than 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several theorems in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.