Definition 1

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

(OM) 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 an ordering which are compatible is called an ordered monoid; if its commutative monoid structure is a group structure, then it is called an ordered group.

In an analogous fashion, we can define the notion of a noncommutative ordered monoid (VI, p. 30, Ex. 1).

If an ordering is compatible with a given monoid structure, then so is the opposite ordering.