Abstract

Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger. In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming that the induced theories on H and are definable complete and o‐minimal respectively, we prove that any definable set of G is ‐definable.