Table of Contents
Residuated partially ordered monoids
Abbreviation: RpoMon
Definition
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 $\cdot$: $x\cdot y\le z\iff x\le z/y$.
Morphisms
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: $x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.
Examples
Basic results
Properties
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.
Finite members
$\begin{array}{lr}
f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\
\end{array}$ $\begin{array}{lr}
f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$
