General Associativity Theorem/Formulation 1

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Theorem

Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left \langle {a_k} \right \rangle_{p+1 \mathop \le k \mathop \le p+n}$ be a sequence of elements of $S$.

Let $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le s}$ be a strictly increasing sequence of natural numbers such that $r_0 = p$ and $r_s = p+n$.


Suppose:

$\displaystyle \forall k \in \left[{1 \,.\,.\, s}\right]: b_k = \prod_{j \mathop = r_{k-1} \mathop + 1}^{r_k} {a_j}$

Then:

$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s {b_k} = \prod_{k \mathop = p \mathop + 1}^{p \mathop + n} {a_k}$


That is:

$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s \left({a_{r_{k-1} + 1} \circ a_{r_{k-1} + 2} \circ \ldots \circ a_{r_k}}\right) = a_{p + 1} \circ \ldots \circ a_{p + n}$


Proof

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

Let $T$ be the set of all $n \in \N_{>0}$ such that:

$(1): \quad$ for every sequence $\left \langle {a_k} \right \rangle_{p+1 \mathop \le k \mathop \le p+n}$ of elements of $S$

and:

$(2): \quad$ for every strictly increasing sequence $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le s}$ of natural numbers such that $r_0 = p$ and $r_s = p+n$:

the statement:

$\displaystyle b_k = \prod_{j \mathop = r_{k-1} \mathop + 1}^{r_k} a_j$

holds.


Basis for the Induction

Let $n = 1$.

Then:

\(\displaystyle n\) \(=\) \(\displaystyle 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle r_s\) \(=\) \(\displaystyle r_0 + 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle s\) \(=\) \(\displaystyle 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle \prod_{k \mathop = 1}^s b_k\) \(=\) \(\displaystyle b_1\)
\(\displaystyle \) \(=\) \(\displaystyle a_{p+1}\)
\(\displaystyle \) \(=\) \(\displaystyle \prod_{k \mathop = p + 1}^{p \mathop + n} a_k\)


So $1 \in T$.

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:

$(1): \quad$ for every sequence $\left \langle {a_k} \right \rangle_{p+1 \mathop \le k \mathop \le p + m}$ of elements of $S$

and:

$(2): \quad$ for every strictly increasing sequence $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le s}$ of natural numbers such that $r_0 = p$ and $r_s = p + m$:

the statement:

$\displaystyle b_k = \prod_{j \mathop = r_{k-1} \mathop + 1}^{r_k} a_j$

holds.


It is to be demonstrated that it follows that:

$(1): \quad$ for every sequence $\left \langle {a_k} \right \rangle_{p+1 \mathop \le k \mathop \le p + m + 1}$ of elements of $S$

and:

$(2): \quad$ for every strictly increasing sequence $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le s}$ of natural numbers such that $r_0 = p$ and $r_s = p + m + 1$:

the statement:

$\displaystyle b_k = \prod_{j \mathop = r_{k-1} \mathop + 1}^{r_k} a_j$

holds.


Induction Step

This is our induction step:

Let $\left \langle {a_k} \right \rangle_{p + 1 \mathop \le k \mathop \le p + m + 1}$ be a sequence of elements of $S$.

Let $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le s}$ be a strictly increasing sequence of natural numbers such that $r_0 = p$ and $r_s = p + m + 1$.

Then $r_{s-1} \le p + m$.

There are two cases:

$(1): \quad r_{s-1} = p + m$
$(2): \quad r_{s-1} < p + m$


First, suppose:

$r_{s-1} = p + m$

Then:

$b_s = a_{p + m + 1}$


Thus:

\(\displaystyle a_{p + 1} \circ \ldots \circ a_{p + m}\) \(=\) \(\displaystyle b_1 \circ \ldots \circ b_{s-1}\) $m \in T$ by Induction Hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle a_{p + 1} \circ \ldots \circ a_{p + m + 1}\) \(=\) \(\displaystyle \left({a_{p + 1} \circ \ldots \circ a_{p + m} }\right) \circ a_{p + m + 1}\) Definition of Composite
\(\displaystyle \) \(=\) \(\displaystyle \left({b_1 \circ \ldots \circ b_{s-1} }\right) \circ b_s\)
\(\displaystyle \) \(=\) \(\displaystyle b_1 \circ \ldots \circ b_s\) Definition of Composite


Secondly, suppose:

$r_{s-1} < p + m$

Let $b\,'_s = a_{r_{s - 1} + 1} \circ \ldots \circ a_{r_s + 1}$.

Then by definition of composite:

$b_s = b\,'_s \circ a_{p + m + 1}$


Now by the Induction Hypothesis:

$m \in T \implies a_{p + 1} \circ \ldots \circ a_{p + m} = b_1 \circ \ldots \circ b_{s-1} \circ b\,'_s$.


Thus:

\(\displaystyle b_1 \circ \ldots \circ b_s\) \(=\) \(\displaystyle \left({b_1 \circ \ldots \circ b_{s-1} }\right) \circ \left({b\,'_s \circ a_{p + m + 1} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({b_1 \circ \ldots \circ b_{s-1} }\right) \circ b\,'_s}\right) \circ a_{p + m + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({b_1 \circ \ldots \circ b_{s-1} \circ b\,'_s}\right) \circ a_{p + m + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({a_{p + 1} \circ \ldots \circ a_{p + m} }\right) \circ a_{p + m + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle a_{p + 1} \circ \ldots \circ a_{p + m} \circ a_{p + m + 1}\)


Thus in both cases $m + 1 \in T$.

So by the Principle of Mathematical Induction:

$T = \N_{>0}$


Hence the result.

$\blacksquare$


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