Abbreviation: PoMon

A \emph{partially ordered monoid} is a structure $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ such that

$\langle A,\cdot,1\rangle$ is a monoid

$\langle G,\le\rangle$ is a partially ordered set

$\cdot$ is \emph{orderpreserving}: $x\le y\Longrightarrow wxz\le wyz$

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

Example 1:

Every monoid with the discrete partial order is a po-monoid.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &4\\
f(3)= &37\\
f(4)= &549\\
f(5)= &\\

\end{array}$

Commutative partially ordered monoids

Lattice-ordered monoids expanded type

Partially ordered semigroups reduced type