Weakly o-minimal structures and Skolem functions

Abstract

The monotonicity theorem is the first step in proving that o-minimal structures satisfy cellular decomposition, which gives a comprehensive picture of the definable subsets in an o-minimal structure. This leads to the fact that any o-minimal structure has an o-minimal theory. We first investigate the possible analogues for monotonicity in a weakly o-minimal structure, and find that having definable Skolem functions and uniform elimination of imaginaries is sufficient to guarantee that a weakly o-minimal theory satisfies one of these, the Finitary Monotonicity Property. In much of the work on weakly o-minimal structures, it is shown that nonvaluational weakly o-minimal structures are most "like" the o-minimal case. To that end, there is a monotonicity theorem and a strong cellular decomposition for nonvaluational weakly o-minimal expansions of a group. In contrast to these results, we show that nonvaluational weakly o-minimal expansions of an o-minimal group do not have definable Skolem functions. As a partial converse, we show that certain valuational expansions of an o-minimal group, called T-immune, do have definable Skolem functions, and we calculate them explicitly via quantifier elimination.