ORDERED O-STABLE GROUPS VIKTOR VERBOVSKIY Abstract. An ordered structure M is called o-λ-stable if for any subset A with |A|≤ λ and for any cut in M there are at most λ 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups. 1. Introduction, notations and preliminaries Since the notion of an o-minimal structure appeared in [9] various generalizations were introduced. Among them are weakly o-minimal structures [6, 8] and quasi-o- minimal theories [3, 4]. It is easy to see that any cut in o-minimal structure has only one completion up to a complete type over the model. For weakly o-minimal structures a similar result has been proved by Kulpeshov [7]: Let M be a totally ordered structure. Then M is weakly o-minimal iff the following holds on M : any cut 〈C,D〉 in M has at most two complete 1-types over M extending 〈C,D〉, and if a cut 〈C,D〉 in M has two complete 1-types over M extending 〈C,D〉, then the set of all realizations of each of these types is convex in any elementary extension of M . It immediately follows from the notion of quasi-o-minimality that each cut has at most continuum extensions up to complete types over a model. What is common for all of these notions. That each cut has a few number of extensions. Recall that stable theories have a few types. So we can combine these things and introduce the notion of an o-stable theory: each cut in each model of this theory has a few complete types which extend it. Let M =(M,<,... ) be a totally ordered structure, a is an element of M and A, B subsets. As usually we write a<A if a<b for any b ∈ A, and A<B if a<b for any a ∈ A and b ∈ B. A partition 〈C,D〉 of M is called a cut if C<D. Given a cut 〈C,D〉 we construct a partial type {c<x<d : c ∈ C, d ∈ D}, which we also call a cut and use the same notation 〈C,D〉. If the set C is definable, then the cut is called quasirational, if in addition sup C ∈ M then the cut 〈C,D〉 is called rational, a non-definable cut is called irrational. If C =(−∞,c) we denote this cut by (c − 0,c), and if C =(−∞,c] we denote it by (c,c + 0). If C = M we denote this cut +∞ and sup A stands for 〈C,D〉, where C = {c ∈ M : c< sup A}. If the set C is definable we will sometimes distinguish cuts defined by sup C and inf D as: sup C stands for 〈C,D〉∪{C(x)} and inf D stands for 〈C,D〉 ∪ {¬C(x)}. A cut 〈C,D〉 in an ordered group is called non-valuational [8, 17] if d − c converges to 0 whenever c and d converge to sup C and inf D accordingly. A cut, which is not non-valuational, is called valuational. Observe that for a valuational cut 〈C,D〉 there is a convex subgroup H such that sup C = sup(a + H) for some a, and this cut is definable iff H is definable. An ordered group G is said to be of non-valuational Date : July 3, 2009. 1