Abbreviation: PoMon

A 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 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