Abstract
We study expansions of (representable) cylindric algebras (R) CA(omega) that are reducts of polyadic equality algebras. The question we address is: Can we add finitely many substitutions, so that in the expanded language of CA(omega), there is a finite set of quantifier free formulas that force representability of the cylindric reduct of algebras satisfying them. A refinement of a known positive solution (with a new proof) is presented. On the other hand, we show that if we expand the language of CA(omega) by finitely many substitutions corresponding to bijections, then in the expanded language, no quantifier free set of formulas containing only finitely many variables axiomatize RCA(omega).