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

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

Let $x, i \in S$.

Suppose that $i$ is idempotent for $\circ$.

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

Then the following statements hold:

$i \mathrel{\mathcal R} x \implies i \mathrel{\mathcal R} x^n$
$x \mathrel{\mathcal R} i \implies x^n \mathrel{\mathcal R} i$


By Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements:

$i \mathrel{\mathcal R} x \implies i^n \mathrel{\mathcal R} x^n$
$x \mathrel{\mathcal R} i \implies x^n \mathrel{\mathcal R} i^n$

By the definition of an idempotent element, $i^n = i$, so the theorem holds.