## Commutative residuated partially ordered monoids

Abbreviation: CRPoMon

### Definition

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

$\cdot$ is commutative: $xy=yx$

Remark: These algebras are also known as 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 $x\le y\Longrightarrow h(x)\le h(y)$.

Example 1:

### 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 quasivariety

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &24\\ f(5)= &131\\ f(6)= &1001\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### References

1) V. de Paiva, 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