Total order compatible with addition on commutative semigroups

In the paper, we describe all total orders $\succ$ compatible with addition on additive subsemigroup $S$ of finite dimensional spaces over rational numbers. We provide a necessary and sufficient condition under which a finitely generated semigroups $S$ equipped with an order $\succ$ is a well-ordered set. We also present some auxiliary results on orders compatible with addition on additive subsemigroups of finite dimensional spaces over real numbers. All arguments in this paper are based on two simple theorems in the geometry of convex (not necessarily closed) sets. Proofs of these theorems are presented for readers’ convenience. A first version of this paper was written as a handout for my graduate course on the theory of Newton–Okounkov bodies.