Powers of Semigroup Element Commute

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: $(4)$