Varying interpolation and amalgamation in polyadic MV-algebras

摘要

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if (21) is the free algebra in the given signature, X_1, X_2 is contained in (21), a is in the subalgebra of (21) generated by X_1, b is in the subalgebra of (21) generated by X_2 and a ≤ b, then there exists an interpolant c in the subalgebra generated by X_1 ∩ X_2 and n ∈ N such that a~n ≤ c ≤ nb. We call this a varying interpolation property because the integer n depends on the inequality a ≤ b. We also address cases where this interpolation property fails, but other weaker (also varying) ones hold. One such interpolation theorem says that though an interpolant c may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to a~n making it smaller and the same number of existential quantifiers are applied to nb making it bigger. This number of quantifiers also varies; it depends on the inequality a ≤ b. Several amalgamation theorems for classes (mostly varieties) of polyadic MV-algebras are obtained. Completeness theorems, relative to Hilbert-style axiomatisations, for the corresponding infinitary many-valued predicate logics are derived using the methodology of algebraic-logic.