Информационная система "Конференции"

Abstracts

Model theory and set theory

An elementary pair of models of a theory T is a structure (M,P) where M is a model of T, and P is a new unary predicate distinguishing an elementary submodel of M. In a joint work with Itay Ben-Yaacov and Anand Pillay, we have generalized Poizat's notion of a "beautiful pair" of stable structures to the more general context of simple structures. The resulting notion of a "lovely pair" of models of a simple theory T is "first order" exactly when T satisfies a combinatorial property (weak nfcp) generalizing Shelah's non-finite cover property (nfcp) to the simple case. In this case, one gets a simple (complete) theory T_P of lovely pairs of models of T. A natural question is how various model theoretic properties of the theory T are reflected in its pair expansion T_P. Among some of the connections found are the following:

Theorem (Vassiliev): For an SU-rank 1 theory T, T_P has finite SU-rank exactly when T is linear (one-based). In this case, for a model (M,P) of T_P, the localization of the geometry of M at P is either trivial or a disjoint union of projective geometries over division rings.

Theorem (Ben-Yaacov, Pillay, Vassiliev): For a countable categorical T (with wnfcp), T_P is countably categorical iff T is one-based.

Applying technique developed in the simple context to the stable case we also prove:

Theorem (Pillay, Vassiliev): Let T be a stable theory with nfcp, and let T_P be the theory of beautiful pairs of models of T. Then T_P eliminaties imaginaries down to imaginaries of T iff there is no infinite group interpretable in T.

A question of describing the "new" imaginaries that appear in the pair expansion is open even in the case of pairs (K,L) of algebraically closed fields. The conjecture is that such imaginaries should be accounted for, in some way, by the elements of G(K)/G(L) where G is an algebraic group defined over the bottom field L.

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