Total order compatible with addition on commutative semigroups

TL;DR

This paper characterizes all total orders compatible with addition on additive subsemigroups of finite-dimensional rational spaces, providing conditions for when such semigroups are well-ordered, based on convex geometry theorems.

Contribution

It offers a complete description of compatible total orders and necessary conditions for semigroups to be well-ordered, using convex geometry techniques.

Findings

Characterization of all total orders compatible with addition.

Necessary and sufficient conditions for semigroups to be well-ordered.

Use of convex set theorems in order theory.

Abstract

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.