Strongly determined types
The paper proposes a weak form of independence for a large class of theories \(T\), including several unstable examples. This independence is based on the notion of strongly determined extension of a type. Given a type \(p\) over \(A\), a strongly determined extension of \(p\) over \(A\) provides, for every model \(M\) of \(T\) containing \(A\), a type \(q\) over \(M\) extending \(p\) such that, if \(\vec{c}\) realizes \(q\), then every partial elementary map from \(M\) to \(M\) fixing \(\text{acl}^{\text{eq}} (A)\) pointwise is elementary over \(\vec{c}\). As said before, strongly determined types suggest an obvious notion of independence. It is noticed that this is just the forking independence in the stable setting, but may differ even for simple unstable theories (in fact, there are simple theories without strongly determined types over \(\emptyset\)). Furthermore, the axioms of symmetry, extension and local character may fail in the new setting. Theories with strongly determined types over any set include some natural unstable cases, such as weakly o-minimal theories. Likewise \(C\)-minimal structures admit strongly determined types, while, for \(P\)-minimal theories, only the existence of a strongly determined type over \(\emptyset\) is proved; for smoothly approximable theories and certain binary homogeneous structures, this existence result is stated over a singleton. On the other hand, some more artificial examples of structures which do not admit strongly determined types over any finite sets are provided. A possible statement of a Finite Equivalence Relation Theorem is discussed. Also developed is a notion of multiplicity for strongly determined types, leading to an analogue of \(\omega\)-stable theories in this setting. \textsl{Involved} finite simple groups in the sense of Hrushovski are treated. Finally, an application to covers of structures is provided.
scientific article; zbMATH DE number 1339985
scientific article