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## Partially ordered groups

Abbreviation: PoGrp

### Definition

A partially ordered group is a structure $\mathbf{G}=\langle G,\cdot,^{-1},1,\le\rangle$ such that

$\langle G,\cdot,^{-1},1\rangle$ is a group

$\langle G,\le\rangle$ is a partially ordered set

$\cdot$ is orderpreserving: $x\le y\Longrightarrow wxz\le wyz$

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

### Examples

Example 1: Any group is a partially ordered group with equality as partial order.

### 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

### 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

Lattice-ordered groups expanded type