In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous theory for arbitrary dependent theories.
We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.
We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
There is much more known about the family of superstable theories when compared to stable theories. This calls for a search of an analogous "super-dependent" characterization in the context of …
There is much more known about the family of superstable theories when compared to stable theories. This calls for a search of an analogous "super-dependent" characterization in the context of dependent theories. This problem has been treated in \cite{Sh:783,Sh:863}, where the candidates "Strongly dependent", "Strongly dependent^2" and others were considered. These families generated new families when we are considering intersections with the stable family. Here, continuing \cite[§2, §5E,F,G]{Sh:863}, we deal with several candidates, defined using dividing properties and related ranks of types. Those candidates are subfamilies of "Strongly dependent". Fulfilling some promises from \cite{Sh:863} in particular \cite[1.4(4)]{Sh:863}, we try to make this self contained within reason by repeating some things from there. More specifically we fulfil some promises from \cite{Sh:863} to to give more details, in particular: in \S4 for \cite[1.4(4)]{Sh:863}, in \S2 for \cite[5.47(2)=Ldw5.35(2)]{Sh:863} and in \S1 for \cite[5.49(2)]{Sh:863}
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
We generalize Frecon's construction of the inevitable radical to groups in stable and even simple theories.
We generalize Frecon's construction of the inevitable radical to groups in stable and even simple theories.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to …
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.
We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a …
We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a simple theory it is the supremum of the weights of all extensions of the type, and a simple theory is strong if and only if all types have finite weight. A theory without the independence property is strong if and only if it is strongly dependent. As a corollary, a stable theory is strongly dependent if and only if all types have finite weight. A strong theory does not have the tree property of the second kind.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.