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