A non representable infinite dimensional quasi-polyadic equality algebra with a representable cylindric reduct

We construct an infinite dimensional quasi-polyadic equality algebra A such that its cylindric reduct is representable, while A itself is not representable. 1 The most well known generic examples of algebraizations of first order logic are Tarski’s cylindric algebras (CA) and Halmos’ polyadic equality algebras (PEA). The theory of cylindric algebras is well developed in the treatise [10], [11], [12] and is still an active part of research in algebraic logic. The generic examples of CA’s are set algebras. More precisely, let α be an ordinal. Let U be a set. Then we define for i, j ∈ α and X ⊆ U : ciX = {s ∈ U : ∃t ∈ X, s(j) = t(j) for all i ̸= j}, dij = {s ∈ U : s(i) = s(j)}. For a set X, let B(X) = ⟨℘(X),∪,∩,∼, ∅, X⟩ be the full Boolean set algebra with universe ℘(X). A cylindric set algebra of dimension α is a subalgebra of an algebra of the form ⟨B(U), ci, dij⟩i,j<α. The class of representable cylindric algebras of dimension α, or RCAα for short, is the class of subdirect products of set algebras of dimension α. These 2000 Mathematics Subject Classification. Primary 03G15.