Abstract

We consider the following generalization of the notion of a structure recursive relative to a set X. A relational structure A is said to be a Γ-structure if for each relation symbol R, the interpretation of R in A is ∑math image relative to X, where β = Γ. We show that a certain, fairly obvious, description of classes ∑math image of recursive infinitary formulas has the property that if A is a Γ-structure and S is a further relation on A, then the following are equivalent: For every isomorphism F from A to a Γ-structure, F is ∑math image relative to X, The relation is defined in A by a ∑math image formula with parameters