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: $x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$.

Examples

Basic results

Properties

Finite members

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

Subclasses

Superclasses

References