We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which (...) may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987. (shrink)

It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every (...) algebra in a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)

We show that it is impossible to define a substitution operator for arbitrary representable cylindric algebras that agrees in its basic properties with the notion of substitutions introduced for dimension complemented algebras.

Let 1 n. We show that the class NrnCAβ of n-dimensional neat reducts of β-dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long-standing open problem of Tarski and his co-authors Andréka, Henkin, Monk and Németi. The proof uses genuine model-theoretic arguments.