### Table of Contents

## Ordered abelian groups

Abbreviation: **OGrp**

### Definition

An \emph{ordered abelian group} is an ordered group $\mathbf{G}=\langle G,+,-,0,\le\rangle$ such that

$+$ is \emph{commutative}: $x+y=y+x$

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 ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x + y)=h(x) + h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

### 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

one-element group

### Subclasses

[[Abelian ordered groups]]

### Superclasses

[[Partially ordered groups]]

[[Ordered monoids]] reduced type