Orbits of subsets of the monster model and geometric theories

par Luis Jaime Corredor

jeudi 4 mai 2023, 16:00 → 17:00 Europe/Paris

112 (ICJ)

Résumé :  Let C be the monster model of a complete first-order theory T.

If D is a subset of C, following D. Zambella we consider  e(D)={D': (C,D)\equiv (C,D')} and o(D)={D': (C,D)\cong (C,D')}$. The general question we ask is when e(D)=o(D) ?  The case where D is A-invariant for some small set A is rather straightforward: it just means that D is definable. We investigate the case where D is not invariant over any small subset. If T is geometric and (C,D) is an H-structure (in the sense of A. Berenstein and E.  Vassiliev) we get some answers. In the case of SU-rank one, e(D)$ is always different from o(D). In the o-minimal case, everything can happen, depending on the complexity of the definable closure.  We also study the case of lovely pairs of geometric theories.