Power Function Preserves Ordering in Ordered Semigroup/Proof 1
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Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.
Let $x, y \in S$ such that $x \preceq y$.
Let $n \in \N_{>0}$ be a strictly positive integer.
Then:
- $x^n \preceq y^n$
where $x^n$ is the $n$th power of $x$.
Proof
By definition of ordered semigroup:
- $\preceq$ is compatible with $\circ$.
By definition of ordering:
- $\preceq$ is transitive.
Thus by Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements:
- $x^n \preceq y^n$
$\blacksquare$