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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:

### Basic results

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

Any finite partially ordered group has only the equality relation as partial order.

### Properties

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ \end{array}$ $\begin{array}{lr} f(6)= &2\\ f(7)= &1\\ f(8)= &5\\ f(9)= &2\\ f(10)= &2\\ \end{array}$

### Subclasses

Abelian partially ordered groups

Lattice-ordered groups expanded type

### Superclasses

Partially ordered monoids reduced type

### References

Trace: » partially_ordered_groups