### Table of Contents

## Commutative integral ordered monoids

Abbreviation: **CIOMon**

### Definition

A \emph{commutative integral ordered monoid} is a commutative (totally) ordered monoid $\mathbf{A}=\langle A,\cdot,1,\le\rangle$ that is

\emph{integral}: $x\le 1$

Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

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

### Definition

A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

### 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)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\

\end{array}$ $\begin{array}{lr}

f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\

\end{array}$

### Subclasses

Abelian ordered groups expansion

### Superclasses

Commutative ordered monoids supervariety

Integral ordered monoids supervariety