Weakly o-minimal nonvaluational structures

A weakly o-minimal structure M=(M,≤,+,…) expanding an ordered group (M,≤,+) is called nonvaluational iff for every cut 〈C,D〉 of (M,≤) definable in M, we have that inf{y−x:x∈C,y∈D}=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 (2000) 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.