A weakly o-minimal structure image expanding an ordered group is called nonvaluational iff for every cut left angle bracketC,Dright-pointing angle bracket of definable in image, we have that inf{y−x:xset membership, variantC,yset membership, variantD}=0. The study of nonvaluational weakly o-minimal expansions of real closed fields carried out in [D. Macpherson, D. Marker, C. Steinhorn,Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 5435–5483. MR1781273 (2001i:03079] suggests that this class is very close to the class of o-minimal expansions of (...) real closed fields. Here we further develop this analogy. We establish an o-minimal style cell decomposition for weakly o-minimal non-valuational expansions of ordered groups. For structures enjoying such a strong cell decomposition we construct a canonical o-minimal extension. Finally, we make attempts towards generalizing the o-minimal Euler characteristic to the class of sets definable in weakly o-minimal structures with the strong cell decomposition property. (shrink)

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets (...) of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y, then for a large set of tuples $\langle \overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$ , where k = dim(Y), the union of fibers $S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}$ is large in Y. Finally, given a weakly o-minimal structure ������, we find various conditions equivalent to the fact that the topological dimension in ������ enjoys the addition property. (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on its (...) value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)