Example 2

The additive group of finite numerical functions defined on a set $E$ is an ordered group under the ordering « $f(x) \leq g(x)$ for all $x \in E$ », written $f \leq g$. This relation says that the graph of the function $f$ lies below that of the function $g$. The reader may find it convenient to think in terms of this graphical interpretation occasionally. \

According to general definitions (Set Theory, IV, p. 264), a bijective map $f$ from an ordered monoid M onto an ordered monoid M' is called an isomorphism of M onto M' if the structure of M' is obtained from that of M by transporting it along $f$. This is equivalent to saying that $f$ is a mapping onto M' such that

(that is to say a homomorphism of the monoid M onto the monoid M'), and that the relations $x \leq y$ and $f(x) \leq f(y)$ are equivalent (whence in particular $f(x) = f(y)$ implies $x = y$, that is $f$ is injective).