Returning to semi-bounded sets

Abstract

An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension 𝒩 such that either 𝒩 is a reduct of an ordered vector space, or there is an o-minimal structure \widehat{𝒩}, with the same universe but of different language from 𝒩, with (i) Every definable set in 𝒩 is definable in \widehat{𝒩}, and (ii) \widehat{𝒩} has an elementary substructure in which every bounded interval admits a definable real closed field. As a result certain questions about definably compact groups can be reduced to either ordered vector spaces or expansions of real closed fields. Using the known results in these two settings, the number of torsion points in definably compact abelian groups in expansions of ordered groups is given. Pillay's Conjecture for such groups follows.