Abbreviation: OMonZ

An \emph{ordered monoid with zero} is of the form $\mathbf{A}=\langle A,\cdot,1,0,\le\rangle$ such that $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ is an ordered monoid and

$0$ is a \emph{zero}: $x\cdot 0 = 0$ and $0\cdot x = 0$

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(0)=0$, $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

$f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &15
f(5)= &84
f(6)= &575
f(7)= &4687
f(8)= &45223
f(9)= &
\end{array}$

Commutative ordered monoids

Ordered monoids reduced type

Ordered semigroups with zero reduced type

Representable residuated lattices