Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements

Theorem
Let $\left({S,\circ}\right)$ be a semigroup.

Let $\mathcal R$ be a transitive relation on $S$ which is compatible with $\circ$.

Let $x,y \in S$, and suppose that $x \mathrel{\mathcal R} y$.

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

Then $x^n \mathrel{\mathcal R} y^n$, where $x^n$ is the $n$th power of $x$.

Proof
We proceed by Mathematical Induction.

By definition of power, $x^1=x$ and $y^1 = y$.

Hence, by assumption, $x^1 \mathrel{\mathcal R} y^1$.

Suppose now that for $n \ge 1$:


 * $x^n \mathrel{\mathcal R} y^n$

Recall the assumption that $x \mathrel{\mathcal R} y$.

Applying Operating on Transitive Relationships Compatible with Operation to these relations yields:


 * $x^n \circ x \mathrel{\mathcal R} y^n \circ y$

By definition of power, $x^{n+1} = x^n \circ x$ and $y^{n+1} = y^n \circ y$.

This completes the induction, and the result follows.