Structure Theorems for o-Minimal Expansions of Groups

In this study the structure for O-minimal expansions of groups is considered Let 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) is semibounded, (2) has no poles, (3) cannot define a real closed field with domain R and order <, (4) is eventually linear and (5) every –definable set is a finit union of cones. As a corollary we get that Th( ) has quantifier elimination and universal axiomatization in the language with symbols for the ordered group operations, bounded –definable sets and symbols for each definable endomorphis of the group (R,0,+).