Powers of Semigroup Element Commute
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Theorem
Let $\struct {S, \odot}$ be a semigroup.
Let $a \in S$.
Let $m, n \in \Z_{>0}$.
Then:
- $\forall m, n \in \Z_{>0}: a^n \odot a^m = a^m \odot a^n$
Proof
\(\ds a^n \odot a^m\) | \(=\) | \(\ds a^{n + m}\) | Index Laws for Semigroup: Sum of Indices | |||||||||||
\(\ds \) | \(=\) | \(\ds a^{m + n}\) | Integer Addition is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds a^m \odot a^n\) | Index Laws for Semigroup: Sum of Indices |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations: $(4)$