Definable sets in ordered structures. III

Abstract

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 <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.

A note on the undecidability of the reachability problem for o-minimal dynamical systems

“…In this paper when we say that some subset or function is definable, we mean it is first-order definable (possibly with parameters) in the sense of the structure M. A general reference for first-order logic is [9]. All the notions related to o-minimality can be found in [17,12,18], see also [19] for a nice overview. We start with the definition of an o-minimal structure: The field of reals R, <, +, ·, 0, 1 , the group of rationals Q, <, +, 0 , the field of reals with exponential function, the field of reals expanded by restricted pfaffian functions and the exponential function are o-minimal structures.…”

“…The o-minimal structures [17,12,18,19] enjoy very nice finiteness properties. Regarding the above discussion it seemed interesting to define "o-minimal transition systems".…”

Filling certain cuts in discrete weakly o-minimal structures

“…For instance, one can observe that, just as in the ominimal framework (see [9]), there is no proper weakly o-minimal expansion of (N , ≤) by "new" (non-definable) functions or relations. In fact, every convex set of (N , ≤) is also an interval (with endpoints in N ∪ {+∞}), so any weakly o-minimal expansion of (N , ≤) is o-minimal and, by [10], also strongly o-minimal. In [9] it was shown that such an expansion must only involve functions and relations which are already definable in (N , ≤).…”

Definable choice for a class of weakly o-minimal theories

“…papers by Knight, Pillay, and Steinhorn (see [10,13,14]); a comprehensive treatment of the subject is also found in the survey text [6]. Throughout this paper, we will be considering expansions of o-minimal groups.…”