## Residuated partially ordered monoids

Abbreviation: RpoMon

### Definition

A residuated partially ordered monoid (or 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: $x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.

### 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.

Classtype order variety 1)

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$