General Commutativity Theorem

== Theorem ==
Let $\struct {S, \circ}$ be a semigroup.
Let $\family {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $S$.
Suppose that:
:$\forall i, j \in \closedint 1 n: a_i \circ a_j = a_j \circ a_i$
Then for every permutation $\sigma: \N_n \to \N_n$:
:$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma n} = a_1 \circ \cdots \circ a_n$
where $\N_n$ is used here to denote the initial segment of $\N_{>0}$:
:$\N_n = \set {1, 2, \ldots, n}$
== Proof ==
The proof will proceed by the Principle of Mathematical Induction on $\N_{>0}$.
Let $T$ be the set of all $n \in \N_{>0}$ such that:
:$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma n} = a_1 \circ \cdots \circ a_n$
holds for all sequences $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ of $n$ elements of $S$ which satisfy:
:$\forall i, j \in \closedint 1 n: a_i \circ a_j = a_j \circ a_i$
for every permutation $\sigma: \N_n \to \N_n$.
=== Basis for the Induction ===
We have that when $n = 1$:
:$\map \sigma 1 = 1$
and so:
:$a_{\map \sigma 1} = a_1$
for all $\sigma: \set 1 \to \set 1$.
Thus $1 \in T$.
This is our basis for the induction.
=== Induction Hypothesis ===
It is to be shown that, if $m \in T$ where $m \ge 1$, then it follows that $m + 1 \in T$.
This is the induction hypothesis:
:$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma m} = a_1 \circ \cdots \circ a_m$
holds for all sequences $\sequence {a_k}_{1 \mathop \le k \mathop \le m}$ of $m$ elements of $S$ which satisfy:
:$\forall i, j \in \closedint 1 m: a_i \circ a_j = a_j \circ a_i$
for every permutation $\sigma: \N^*_m \to \N^*_m$.
It is to be demonstrated that it follows that:
:$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma {m + 1} } = a_1 \circ \cdots \circ a_{m + 1}$
holds for all sequences $\sequence {a_k}_{1 \mathop \le k \mathop \le m + 1}$ of $m + 1$ elements of $S$ which satisfy:
:$\forall i, j \in \closedint 1 {m + 1}: a_i \circ a_j = a_j \circ a_i$
for every permutation $\sigma: \N_{m + 1} \to \N_{m + 1}$.
=== Induction Step ===
This is our induction step:
Suppose $m \in T$.
Let $\sequence {a_k}_{1 \le k \le m + 1}$ be a sequence of $m+1$ elements of $S$ which satisfy:
:$\forall i, j \in \closedint 1 {n + 1}: a_i \circ a_j = a_j \circ a_i$
Let $\sigma: \N_{m + 1} \to \N_{m + 1}$ be a permutation of $\closedint {1 {m + 1}$.
There are three cases to consider:
:$(1): \quad \map \sigma {m + 1} = m + 1$
:$(2): \quad \map \sigma 1 = m + 1$
:$(3): \quad \map \sigma r = m + 1$ for some $r \in \closedint 2 m$
$(1): \quad$ Suppose $\map \sigma {m + 1} = m + 1$.
Then the restriction of $\sigma$ to $\N_m$ is then a permutation of $\N_m$.
From the induction hypothesis:
:$m \in T$:
Thus:
:$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma m} = a_1 \circ \cdots \circ a_m$
from which:
$(2): \quad$ Suppose $\map \sigma 1 = m + 1$.
Let $\tau: \N^*_m \to \N^*_m$ be the mapping defined as:
:$\forall k \in \closedint 1 m: \map \tau k = \map \sigma {k + 1}$
From Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor:
:$\closedint 1 {m + 1} = \closedint 1 m \cup \set {m + 1}$
Thus $\tau$ is clearly a permutation on $\closedint 1 m$.
Thus, as $m \in T$:
:$a_{\map \tau 1} \circ \cdots \circ a_{\map \tau m} = a_1 \circ \cdots \circ a_m$
So:
$(3): \quad$ Suppose $\map \sigma r = m + 1$ for some $r \in \closedint 2 m$.
Let $\tau: \N_{m + 1} \to \N_{m + 1}$ be a mapping defined by:
:$\forall k \in \N_{m + 1}: \map \tau k = \begin{cases}
\map \sigma k & : k \in \closedint 1 {r - 1} \\
\map \sigma {k + 1} & : k \in \closedint r m \\
m + 1 & : k = m + 1 \end{cases}$
It is seen that $\tau$ is a permutation of $\N_{m + 1}$.
So, by the first case:
:$a_{\map \tau 1} \circ \cdots \circ a_{\map \tau {m + 1} } = a_1 \circ \cdots \circ a_{m + 1}$
Thus:
So in all cases, $m + 1 \in T$.
Thus by the Principle of Mathematical Induction:
:$T = \N_{>0}$
The result follows.
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations: Theorem $18.7$
* : $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Corollary $1.7$
Category:Named Theorems
Category:Commutativity