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.