  Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an
expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we
prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff
it has the exchange property and any bounded definable set has a maximum.
