### Definable sets in ordered structures. III

Anand Pillay, Charles Steinhorn
1988 Transactions of the American Mathematical Society
We show that any o-minimal structure has a strongly o-minimal theory. 0. 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
more » ... tions 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 <m is a linear ordering, and every definable (with parameters) subset X c M is a finite union of points and intervals (a, b) (where a E M U {-co}, b E M U {oo}). We use freely notation and results from previous papers on the subject [1, 2 and 3]. 1. The discrete case. We say that the o-minimal structure M is discrete if every element a of M has an immediate successor S(a) and an immediate predecessor S~1(a). This is rather a strong definition, and our results here are valid for M satisfying a weaker notion of discrete, as we subsequently point out. We now fix discrete o-minimal M. DEFINITION 1.1. Let X c M. We say that X is scattered if for no a E M does X contain both a and S(a). Note that by o-minimality, any definable scattered X c M is finite. We are going to prove THEOREM 1.2. Let <p(x, y) E L(M) be such that, for every a C M (1(a) = l(x)), <p(a,y)M is scattered. Then there is N < uj such that, for every a, \<p(a,y)M\ < N. This will be proved by induction on n = l(x). First we need some preliminary definitions.