This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers and $\mathcal{R} = (\mathbb{R}\cup\{\infty\},\min,+)$ the tropical semiring. Let $\mathbb{Z}_n$ denote the cyclic group of order $n$.