CAYLEY’S THEOREM FOR ORDERED GROUPS: O-MINIMALITY

It has long been known [4] that any group could be represented in a strongly minimal theory by just writing down the relations of the group as unary functions. We show the same process works for ordered groups and yields an ominimal group. Baizanhov showed: Fact 1. If M is an o-minimal structure and p ∈ S(A) is non-algebraic then there is an ordered group of unary definable functions defined on the realizations of p. He asked, Question 2. What ordered groups can be realized in this manner? We show Theorem 3. If G is a linearly ordered group, then G can be represented as the collection of definable unary functions acting on a complete type in an 0-minimal structure. Proof of Fact 1. This follows from a short analysis of p−stable formulas [1]. A 2-ary formula, φ(x, y), over A is p−stable if for some (equivalently for every) realization α ∈ p(M) there exist γ1, γ2 ∈ p(M) such that γ1 < φ(M, α) < γ2. (In particular, x = y is such a 2-formula.) By o-minimality, φ(M, α), can be written as a finite number of L(Aα)definable intervals and points. Thus each p-stable formula determines a finite number of functions: fi maps α to the ith point in a linear ordering of the end points of these intervals. Let f be such function. The image f(α) and pre-image f −1 (α) of an arbitrary element α ∈ p(M) belong to p(M). We can define f −1, becaus