$a,b,c\in \mathcal{M}$. An ordered monoid is commutative if $+$ is commutative.

A tuple $(\mathcal{M},+,-,\le)$, where $(\mathcal{M},+,\leq)$ is an ordered monoid and $-$ is a binary operation on $\mathcal{M}$, is called an *ordered monoid with subtraction*.

We say that subtraction is *well-behaved* whenever for all $a,b\in \mathcal{M}$ it holds that $a-a = 0$ and $(a-b)+b = a$ whenever $b\le a$.

For now subtraction is just any operation, without specific requirements. Later we will concentrate on specific subtraction operations and demand that they are well-behaved.

In the following we will consider only commutative monoids.

Let $\mathcal{M}_1$, $\mathcal{M}_2$ be two ordered monoids. A map $h\colon \mathcal{M}_1\to \mathcal{M}_2$ is called *monotone* if $a\le b$ implies $h(a)\le h(b)$ for all $a,b\in
  \mathcal{M}_1$.

$h$ is called a *homomorphism* if $h(0)=0$ and $h(a+b)= h(a)+h(b)$. If $\mathcal{M}_1,\mathcal{M}_2$ are ordered monoids with subtraction, we say that $h$ preserves subtraction if $h(a-b) = h(a)-h(b)$.

\[ex:counting-monoid\] Let $n \in \mathbb{N}\backslash\{0\}$ and take $\mathcal{M}_n = \{0,1,\dots,n,*\}$ (zero, one, $\dots$, $n$, many) with $0 \leq 1 \leq \dots\leq n\leq *$ and addition as (commutative) monoid operation with the proviso that $a+b=*$ if the sum is larger than $n$. In addition $a+* = *$ for all $a\in
  \mathcal{M}_n$. Subtraction is truncated subtraction where $a-b 