Power of Element/Semigroup

Theorem
Let $\struct {S, \oplus}$ be a magma.

Let $a \in S$.

Let $n \in \N_{>0}$.

Let $\tuple {a_1, a_2, \ldots, a_n}$ be the ordered $n$-tuple defined by $a_k = a$ for each $k \in \N_n$.

Then:
 * $\displaystyle \bigoplus_{k \mathop = 1}^n a_k = \oplus^n a$

where:
 * $\displaystyle \bigoplus_{k \mathop = 1}^n a_k$ is the composite of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$
 * $\oplus^n a$ is the $n$th power of $a$ under $\oplus$.

Proof
The proof will proceed by the Principle of Mathematical Induction on $\N$.

Let $T$ be the set defined as:
 * $\displaystyle T := \set {n \in \N: \bigoplus_{k \mathop = 1}^n a_k = \oplus^n a}$

First, recall the definition of the composite of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$:


 * $\displaystyle \bigoplus_{k \mathop = 1}^n a_k = \begin{cases}

a: & n = 1 \\ \map {\oplus_m} {a_1, \ldots, a_m} \oplus a_{m + 1}: & n = m + 1 \end{cases}$

Secondly, recall the definition of the $n$th power of $a$ under $\oplus$:


 * $\forall n \in \N_{>0}: \oplus^n a = \begin{cases}

a & : n = 1 \\ \paren {\oplus^m a} \oplus a & : n = m + 1 \end{cases}$

Basis for the Induction
We have that:

So $1 \in T$.

This is our basis for the induction.

Induction Hypothesis
It is to be shown that, if $j \in T$ where $j \ge 1$, then it follows that $j + 1 \in T$.

This is the induction hypothesis:
 * $\displaystyle \bigoplus_{k \mathop = 1}^j a_k = \oplus^j a$

It is to be demonstrated that it follows that:
 * $\displaystyle \bigoplus_{k \mathop = 1}^{j + 1} a_k = \oplus^{j + 1} a$

Induction Step
This is our induction step:

So $k \in T \implies k + 1 \in T$ and the result follows by the Principle of Mathematical Induction:


 * $\displaystyle \forall n \in \N: \bigoplus_{k \mathop = 1}^n a_k = \oplus^n a$