A partial type $ \Sigma(x) $, over some set of parameters, is said to be stable if every extension to a complete type $ p(x) $ over a model $ M $ is $ M $-definable. Here, a "model" is an elementary extension of the original structure, or, in the context of a monster model, an elementary substructure of a model.

Stable types are also called fully stable types, or stable and stably embedded sets. The set of realizations of a stable type is always stably embedded.

Any type of Morley rank less than $ \infty $ is a stable type (right?). In a stable theory, all types are stable, and conversely if all types are stable, then the ambient theory is stable.

Stable types can alternatively be characterized by counting types, or by the order property. For example, $ \Sigma(x) $ is unstable if and only if there is a formula $ \phi(x;y) $, a sequence $ \langle a_n,b_n \rangle_{n < \omega} $ with $ \Sigma(a_n) $ holding, such that $ \models \phi(a_i,b_j) $ if and only if $ i < j $.

References Edit

Stable types were apparently introduced by Lascar and Poizat. They are discussed in Poizat's book on Model Theory.

A number of basic facts and observations around stable types are given in Onshuus and Usvyatsov's paper Stable Domination and Weight.

The appendix to Chatzidakis and Hrushovski's Model Theory of Difference Fields briefly discusses stable types.