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.