## 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}: $x+y=y+x$

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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 …: $axiom$

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 universal

### Finite members

one-element group

### Subclasses

[[Abelian ordered groups]]

### Superclasses

[[Partially ordered groups]]
[[Ordered monoids]] reduced type