Set Theory and Topology seminar (BGU)
A linearly ordered structure (M,<,…) is weakly o-minimal if every definable subset of the line is a finite union of convex sets. It is o-minimal if every definable subset of the line is a finite union of intervals. A weakly o-minimal expansion of a group is non-valuational if every definable cut has 0 diameter (the canonical example of a valuational weakly o-minimal strcture being a real closed valued field, where the cut determined by the valuation ring has infinite diameter). By a theorem of Wencel, every weakly o-minimal non-valuational expansion of a group M embeds canonically as a dense sub-set of an o-minimal structutre M’, and the structure M’ induces on (the universe of) M is precisely M.
By a theorem of Poizat and Baizalov, if M is an o-minimal structure, N a proper elementary sub-structure then the structure induced on N by all M-definable sets (namely, all sets of the form S\cap N^n where S is an M^definable subset of M^n) is weakly o-minimal. In case M expands a group, and N is dense in M the resulting o-minimal structure is weakly o-minimal and non-valuational. We call a structure obtained in this way an o-minimal trace. The canonical example for such a structure is the field of real algebraic numbers expanded by a unary predicate for the cut at \pi.
This is part of a paper to appear jointly with P. Elftheriou and G. Keren.