Structure theorems for o-minimal expansions of groups Original Research Article

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