Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements
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Theorem
Let $\left({S, \circ}\right)$ be a semigroup.
Let $\mathcal R$ be a transitive relation on $S$ which is compatible with $\circ$.
Let $x, y \in S$ such 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:
- $x^1 = x$
- $y^1 = y$
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:
- $y^{n+1} = y^n \circ y$
The result follows by the Principle of Mathematical Induction.
$\blacksquare$