A note on substitutions in representable cylindric algebras

Abstract

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.

Existence of partial transposition means representability in cylindric algebras

“…We generalize (14) [7], [5], [10] and [11] are related to the above Theorem. [7, Theorem 3.2.52] includes a sufficient condition for A to be a reduct of a quasi polyadic algebra.…”

“…So (17) is true. By(11), M ∈ PTA α+1 . So(19) [i, j] x = k S(i, j)x if Δx ⊆ α, k ≤ α, by (12) in the proof of Theorem 3.9.…”

On the definition and the representability of quasi‐polyadic equality algebras

“…Regarding another important algebraization of first order logic, cylindric algebras, a cylindric algebra cannot always have an associated quasi-polyadic algebra, because the substitution operator s τ cannot always be introduced in cylindric algebras (cf. [4,16]). Therefore, quasi-polyadic equality algebras are said to be between polyadic algebras and cylindric algebras, or they can be considered as cylindric algebras endowed with a substitution operator s τ , where τ is finite.…”