Properties of Ordered Group/OG5/Proof 1

Theorem
Let $\struct {G, \circ, \preceq}$ be an ordered group with identity $e$.

Let $\prec$ be the reflexive reduction of $\preceq$.

Let $x \in G$.

Let $n \in \N_{>0}$ be a strictly positive integer.

Then the following hold:


 * $x \preceq e \implies x^n \preceq e$
 * $e \preceq x \implies e \preceq x^n$
 * $x \prec e \implies x^n \prec e$
 * $e \prec x \implies e \prec x^n$

Proof
By Power Function Preserves Ordering in Ordered Group:
 * $x \preceq e \implies x^n \preceq e^n$
 * $e \preceq x \implies e^n \preceq x^n$
 * $x \prec e \implies x^n \prec e^n$
 * $e \prec x \implies e^n \prec x^n$

By Identity Element is Idempotent, $e$ is idempotent with respect to $\circ$.

Therefore by the definition of an idempotent element, $e^n = e$.

Thus the theorem holds.