Abbreviation: RpoMon
A \emph{residuated partially ordered monoid} (or \emph{rpo-monoid}) is a structure A=⟨A,≤,⋅,1,∖,/⟩ such that
⟨A,≤⟩ is a partially ordered set,
⟨A,⋅,1⟩ is a monoid and
∖ is the left residual of ⋅: x⋅y≤z⟺y≤x∖z
/ is the right residual of ⋅: x⋅y≤z⟺x≤z/y.
Let A and B be residuated po-monoids. A morphism from A to B is a function h:A→B that is an order-preserving homomorphism: x≤y⟹h(x)≤h(y), h(x⋅y)=h(x)⋅h(y), h(x∖y)=h(x)∖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.
$\begin{array}{lr}
f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\
\end{array}\begin{array}{lr}
f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$