Definition 2

— A pre-order relation (written $\leq$) and a commutative monoid structure (written additively) on a set $M$ are said to be compatible if they satisfy the following axiom:

(POM) For each $z \in M$, the relation $x \leq y$ implies $x + z \leq y + z$.

A set $M$ equipped with a commutative monoid structure and a pre-order relation which are compatible is called a pre-ordered monoid.

Let $M$ be a pre-ordered monoid, and $S$ the equivalence relation « $x \leq y$ and $y \leq x$ ». By virtue of (POM) the relation $x \equiv x' \pmod{S}$ implies $x + y \leq x' + y$ and $x' + y \leq x + y$ for all $y \in M$, that is $x + y \equiv x' + y \pmod{S}$. In other words the equivalence relation $S$ is compatible with addition in $M$ (I, p. 11). Thus the quotient by $S$ of the addition law on $M$, together with the ordering associated to $\leq$, gives $M/S$ the structure of an ordered monoid. In the case where $M$ is a pre-ordered group, the group $M/S$ is the quotient of $M$ by the subgroup consisting of all elements $x$ such that $x \leq 0$ and $0 \leq x$.