On \(\aleph_0\)-categorical weakly o-minimal structures
A structure \(M\) is o-minimal, if it carries a linear order such that every definable subset of \(M\) is a finite union of intervals of this order. A complete classification of \(\aleph_0\)-categorical, o-minimal structures was given by \textit{A. Pillay} and \textit{C. Steinhorn} [``Definable sets in ordered structures'', Bull. Am. Math. Soc., New Ser. 11, 159-162 (1984; Zbl 0542.03016)]. The paper under review continues this work to give a classification of \(\aleph_0\)-categorical, weakly o-minimal structures. Call a subset \(C\) of \(M\) convex, if for any two points \(x, y \in C\) all points between \(x\) and \(y\) are also contained in \(C\). Then \(M\) is called weakly o-minimal, if every definable subset is a finite union of convex subsets. The authors consider only 1-indiscernible structures, i.e. structures with a single 1-type. They show that up to binary structure, \(Q^n\) ordered lexicographically and equipped with the metric \(d: Q^n \times Q^n \to \{0,\ldots, n\}\) described below, is essentially the only \(\aleph_0\)-categorical, weakly o-minimal structure with \(n\) 2-types. The metric \(d\) is given by \(d(x_0,\dots{},x_{n-1},y_0,\dots{},y_{n-1})=k\) iff \(x_k \neq y_k\) and for all \(j<k\) \(x_j = y_j\). The metric \(d\) is an ultrametric in the sense that \(d(x,z) \leq\max(d(x,y), d(y,z))\) for all \(x,y,z\). For 2-indiscernible structures the classification is carried out in a similar spirit up to ternary structure, using the concept of C-relations. It turns out that the \(\aleph_0\)-categorical, weakly o-minimal structures with one 2-type and exactly \(n\) 3-types admit an axiomatisation in terms of ultrametrics. Structures that are 3-indiscernible are in fact \(k\)-indiscernible for every finite \(k\) and hence essentially isomorphic to the ordered set of the rationals, \((Q,<)\). The picture is rounded by a number of examples and considering relations of higher arities.
scientific article; zbMATH DE number 1398638