Abbreviation: CRPoMon

A \emph{commutative residuated partially ordered monoid} is a residuated partially ordered monoid $\mathbf{A}=\langle A, \cdot, 1, \to, \le\rangle$ such that

$\cdot$ is \emph{commutative}: $xy=yx$

Remark: These algebras are also known as \emph{lineales}.¹⁾

Let $\mathbf{A}$ and $\mathbf{B}$ be commutative residuated partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(x \to y)=h(x) \to h(y)$, and $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

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)= &2\\
f(3)= &5\\
f(4)= &24\\
f(5)= &131\\
f(6)= &1001\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Commutative residuated lattices expansion

Pocrims same type

Residuated partially ordered monoids supervariety

Commutative partially ordered monoids subreduct