  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.
