Submission date: 7 February 2017

Abstract:

Let M be the monster model of a complete first-order theory T. If D is a subset of M, following D. Zambella we consider e(D)={ D' | (M,D) ≡ (M,D') } and o(D)={ D' | (M,D) ≅ (M,D') }. The general question we ask is when e(D)=o(D)? The case where D is A-invariant for some small set A is rather straightforward: it just mean that D is definable. We investigate the case where D is not invariant over any small subset. If T is geometric and (M,D) is an H-structure (in the sense of A. Berenstein and E. Vassiliev) or a lovely pair, we get some answers. In the case of SU-rank one, e(D) is always different from o(D). In the o-minimal case, everything can happen, depending on the complexity of the definable closure.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 1702.01892: pdf, ps.