ABSTRACT. We show that any o-minimal structure has a strongly o-minimal theory.
Introduction.In this paper we prove that an arbitrary o-minimal structure M is strongly o-minimal. This was proved in [1] in the case when the ordering on M is dense.In §1 we show that for discrete M, o-minimal implies strongly o-minimal. This is, of course, a result on uniform finite bounds. The proof has some interesting differences with the dense case, partly because here one has to prove uniform bound results for functions defined on finite intervals. In [3] it was shown that strongly ominimal discrete structures are "trivial", i.e. there are no definable functions other than translations in one variable. Thus, with the results of §1, discrete o-minimal structures lose their interest.In §2 we show that the discrete and dense parts of an arbitrary o-minimal structure are "orthogonal", from which our main result follows.Recall that the structure (M, <,...) is said to be o-minimal if