Subsets of the monster model and geometric theories
Speaker: Enrique Casanovas
Abstract: This is a joint work with L.J. Corredor. 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), the collection of all D' such that (M,D) and (M,D') are elementarily equivalent and o(D), the collection of all D' such that (M,D) and (M,D') are isomorphic. The general question we ask is when e(D)=o(D). It is rather straightforward to find the answer if D is A-invariant for some small set A: 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.