Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \circ}$ be a semigroup.
Let $\RR$ be a transitive relation on $S$ which is compatible with $\circ$.
Let $x, y \in S$ such that $x \mathrel \RR y$.
Let $n \in \N_{>0}$ be a strictly positive integer.
Then:
- $x^n \mathrel \RR y^n$
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:
- $x^1 \mathrel \RR y^1$
Suppose now that for $n \ge 1$:
- $x^n \mathrel \RR y^n$
Recall the assumption that $x \mathrel \RR y$.
Applying Operating on Transitive Relationships Compatible with Operation to these relations yields:
- $x^n \circ x \mathrel \RR y^n \circ y$
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.
$\blacksquare$