Elimination of imaginaries

Abstract

Let A be a structure for a language L. We already dealt with definability in A when we introduced definable sets and, more generally, definable structures in A (see example 1.7.3, 4). The latter are the structures A′ for a language L′ (possibly different from L) such that both the universe A′ of A′ and the interpretations of symbols of L′ in A′ are definable in A. As in the case of sets, we can introduce also the concept of X-definable structure for X⊆A. In the quoted example, we observed that (N, +, ·) is a structure definable in (Z, +, ·). Here we provide some further examples.