Fonctionne avec Indico
v3.2.9
A totally ordered structure is said to be λ-o-stable if for any subset A of cardinality at most λ and any cut s over the structure there are at most λ 1-types over the set A which are consistent with s. A theory is λ-o-stable if all its models are. A theory is o-stable if it is λ-o-stable for some infinite λ. I will consider ordered groups and fields with an o-stable theory. In this fields any infinite definable subset contains a non-empty interior and, as a corollary, is a union of an open set and a finite set. Each definable unary function is piecewise monotone, where pieces are convex, but the number of them can be infinite. Any o-stable ordered field is real closed. Also I will give a characterisation of pure ordered o-stable groups.