Power Function Preserves Ordering in Ordered Group
Theorem
Let $\struct {S, \circ, \preccurlyeq}$ be an ordered group.
Let $n \in \N_{>0}$ be a strictly positive integer.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Then the following hold:
| \(\ds \forall x, y \in S: \, \) | \(\ds x \preccurlyeq y\) | \(\implies\) | \(\ds x^n \preccurlyeq y^n\) | |||||||||||
| \(\ds \forall x, y \in S: \, \) | \(\ds x \prec y\) | \(\implies\) | \(\ds x^n \prec y^n\) |
where $x^n$ denotes the $n$th power of $x$.
Corollary
| \(\ds \forall x \in S: \, \) | \(\ds x \preccurlyeq e\) | \(\implies\) | \(\ds x^n \preccurlyeq e\) | |||||||||||
| \(\ds e \preccurlyeq x\) | \(\implies\) | \(\ds e \preccurlyeq x^n\) | ||||||||||||
| \(\ds x \prec e\) | \(\implies\) | \(\ds x^n \prec e\) | ||||||||||||
| \(\ds e \prec x\) | \(\implies\) | \(\ds e \prec x^n\) |
Proof 1
By definition of ordered group:
- $\preccurlyeq$ is compatible with $\circ$.
By definition of ordering:
- $\preccurlyeq$ is transitive.
From Reflexive Reduction of Relation Compatible with Group Operation is Compatible:
- $\prec$ is also compatible with $\circ$.
From Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering:
- $\prec$ is also transitive.
By definition of ordered group:
- $\struct {S, \circ}$ is a ordered group, and therefore a fortiori a semigroup.
The result follows from Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements.
$\blacksquare$
Proof 2
An ordered group is an ordered structure which is also a group.
Hence an ordered group is a fortiori an ordered semigroup.
From Power Function Preserves Ordering in Ordered Semigroup:
- $\forall x, y \in S: x \preccurlyeq y \implies x^n \preccurlyeq y^n$
From the Cancellation Laws, every element of a group is cancellable.
Hence from Power Function with Cancellable Element Preserves Strict Ordering in Ordered Semigroup:
- $\forall x, y \in S: x \prec y \implies x^n \prec y^n$
$\blacksquare$