An interpolation theorem

Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of rst-order logic, there is an interpolant in which each relation symbol appears only in those polarities in which it appears in both the antecedent and the succedent of the given implication. This note proves a similar interpolation result under the additional requirement that, for some xed tuple U of unary predicates U, all formulae under consideration have all quantiiers explicitly relativized to one of the U. Under this stipulation, existential / universal quantiication over U contributes a positive / negative occurrence of U. It is shown how this single new interpolation theorem uniies a number of related results: namely, the classical preservation theorems concerning extensions / substructures with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on existentially / universally quantiied sorts. Consider rst-order logic with or without equality in a nite, purely relational vocabulary. Boolean constants > and ? are taken to be atomic rst-order formulae. We use the set Occ = f+; ?g to code polarities of predicate occurrences in formulae, through a