Structure theorems for o-minimal expansions of groups (Q1964148)

Let \(R\) be an o-minimal expansion of an ordered group \((R;0,1,+,<)\) with distinguished positive element \(1\). The following statements are equivalent: (1) \(R\) is semi-bounded; (2) \(R\) has no poles; (3) \(R\) cannot define a real closed field with domain \(R\) and order \(<\); (4) \(R\) is eventually linear; (5) every \(R\)-definable set is a finite union of cones. As a corollary, \(\text{Th}(R)\) has quantifier elimination and universal axiomatization in the language with symbols for ordered group operations, bounded \(R\)-definable sets and a symbol for each definable endomorphism of the group \((R;0,+)\).