\begin{definition}[Definition:Meaningful Product]
Let $\left({S, \circ}\right)$ be a semigroup.
Let $a_1, \ldots, a_n$ be a sequence of elements of $S$.
Then we define a '''meaningful product''' of $a_1, \ldots, a_n$ inductively as follows:
If $n = 1$ then the only meaningful product is $a_1$.
If $n > 1$ then a meaningful product is defined to be any product of the form:
:$\left({a_1 \ldots a_m}\right)\left({a_{m+1} \ldots a_n}\right)$
where $m < n$ and $\left({a_1 \ldots a_m}\right)$ and 
$\left({a_{m+1} \ldots a_n}\right)$ are meaningful products of $m$ and $n - m$ elements respectively.
\end{definition}