### 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  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  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.