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

Abstract

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (QPENα), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in QPENα are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in Gwqα.

On the algebraization of Henkin‐type second‐order logic

“…We can conclude that if in (8) the original definition of Henkin system is replaced by the equality in (10), then the same algebra is obtained. Thus the following corollary is true: Corollary 2.4 Theorem 2.1 remains true if the definition of Henkin system {A n } n∈ω in (8) is changed by the property in (10). (10) and Proposition 2.3 allow us to give an even abstract characterization of the algebraic version of Henkin model.…”

“…Thus the following corollary is true: Corollary 2.4 Theorem 2.1 remains true if the definition of Henkin system {A n } n∈ω in (8) is changed by the property in (10). (10) and Proposition 2.3 allow us to give an even abstract characterization of the algebraic version of Henkin model. This is obtained by the mix of the above two approaches of the algebraization.…”

“…The point in the proof of the above proposition is that the closure properties (i)-(vii) imply that C consists of definable sets in L C . Since A n ⊂ D, the critical inclusion in (10) (10) implies that A n is definable on the minimal language, i.e., on the language L 2 of the frame.…”

“…Therefore the algebra A in (8) and F coincide. ( 12) is satisfied in F because (10) is transformed from the Henkin model by the transformations in § § 2.1 and 2.2.…”

“…i j = {y ∈ V : y i = y j }, and A is closed under the operations C V i and includes the diagonals D V i j (cf. [2,9,10,15]). Here y i u denotes the element of U ω for which y i (y ∈ U ω ) is replaced by u ∈ U. U is the base of A, V is the unit of A and A is the universe.…”