The addition on totally positive integers uniquely determines the totally real number field

Let $K$ be a totally real number field. We prove that the totally positive algebraic integers $\mathcal O_K^+$ of $K$, viewed as an abstract additive semigroup, uniquely determine $K$. We also describe all the additive relations by giving a presentation of $\mathcal O_K^+$ in terms of indecomposables as generators and a certain concrete set of generating relations. In fact, we obtain this presentation in a more general class of discrete semigroups which we call semilattices.