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 , with the same universe but of different language from N, with (i) Every definable set in N is definable in , and (ii) 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.

A note on generic subsets of definable groups

“…We end the section with an observation about another uniform definability result from [14]. In [14,Lemma 7.4 (ii)] it is proved that in an o-minimal expansion of an ordered group, if {G s : s ∈ S} is a uniformly definable family of abelian definable groups, then the set of s for which G s is definably connected is definable.…”

“…Pillay's conjecture has now been proved in three different situations: in ominimal expansions of fields by Hrushovski, Peterzil and Pillay [13], in linear ominimal expansions of ordered groups by Eleftheriou and Starchenko [12] and in non-linear semi-bounded o-minimal expansions of groups by Peterzil [14]. So Pillay's conjecture holds in arbitrary o-minimal expansions of groups.…”

“…So Pillay's conjecture holds in arbitrary o-minimal expansions of groups. In all of the three cases above the conjecture is a consequence of the following two crucial ingredients (after the paper [1] by Berarducci, Otero, Peterzil and Pillay where the existence of G 00 with properties (1) and (2) is proved): (i) the model theory of generic subsets of definably compact definable groups from [17] together with the heavy model theory of definable ameanable groups from [13]; (ii) the computation of m-torsion subgroups of abelian definably compact definable groups ( [7], [12] and [14] respectively in each one of the three cases).…”

“…The computation of m-torsion subgroups of abelian definably compact definable groups in the field case uses the o-minimal singular cohomology theory from [10] and [22], in the linear case this is obtained by a structure theorem for such definable groups ( [12]) and in the non-linear semibounded case by reduction to the field case ( [14]). With a good cohomology theory in arbitrary o-minimal structures which generalizes the o-minimal singular cohomology in o-minimal expansion of real closed fields ( [10] and [22]) one could obtain a uniform proof of the computation of m-torsion subgroups of abelian definably compact definable groups in arbitrary o-minimal structures which would include the three cases above.…”

Definable group extensions in semi‐bounded o‐minimal structures

“…The climax of these results was Pillay's Conjecture for definably compact groups from [Pi2] and its solution in case R is linear ( [ElSt]) and in case R expands a real closed field ( [HPP]). More recently, the conjecture was solved in the intermediate case where R is semi-bounded ( [Pet2]), by a reduction of the conjecture to the field case. This concluded the proof of Pillay's Conjecture in arbitrary o-minimal expansions of ordered groups.…”