=====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}.[(dePaiva2005)]

==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.

^[[Classtype]]                        |quasivariety  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

[(dePaiva2005>
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]])]