Abbreviation: RpoMon

A \emph{residuated partially ordered monoid} (or \emph{rpo-monoid}) is a structure $\mathbf{A}=\langle A,\le,\cdot,1,\backslash,/\rangle$ such that

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

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

$\backslash$ is the left residual of $\cdot$: $x\cdot y\le z\iff y\le x\backslash z$

$/$ is the right residual of $\cdot$: $x\cdot y\le z\iff x\le z/y$.

Let $\mathbf{A}$ and $\mathbf{B}$ be residuated po-monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an order-preserving homomorphism: $x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.

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.

$\begin{array}{lr}

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

\end{array}\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Commutative residuated partially ordered monoids

Involutive residuated partially ordered monoids

Residuated lattices

Partially ordered monoids

Residuated partially ordered semigroups