Assaf Hasson: On weakly o-minimal structures and o-minimal traces.

Set Theory and Topology seminar (BGU)

Time: Tuesday, April 21st, 12:15-13:40.
Place: Seminar room -101, Math building 58.
Speaker: Assaf Hasson (BGU).
Title: On weakly o-minimal structures and o-minimal traces.

Abstract:
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.

In view of Wencel’s theorem it is natural to ask: are all weakly o-minimal non-valuational expansions of groups o-minimal traces. In the talk we will present examples showing that reducts of o-minimal traces need not be o-minimal reducts. We will also present examples of weakly o-minimal non-valuational structures that are not reducts of o-minimal traces.

This is part of a paper to appear jointly with P. Elftheriou and G. Keren.

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