### Table of Contents

## Commutative residuated partially ordered monoids

Abbreviation: **CRPoMon**

### Definition

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)}

##### Morphisms

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 $x\le y\Longrightarrow h(x)\le h(y)$.

### Examples

Example 1:

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

\end{array}$

### Subclasses

Commutative residuated lattices expansion

Pocrims same type

### Superclasses

Residuated partially ordered monoids supervariety

Commutative partially ordered monoids subreduct

### References

^{1)}V. de Paiva, \emph{Lineales: Algebras and Categories in the Semantics of Linear Logic}, Proofs and Diagrams, CSLI Publications, Stanford, 123-142, 2005, https://research.nuance.com/wp-content/uploads/2014/10/Lineales-algebras-and-categories-in-the-semantics-of-Linear-Logic.pdf