Definition:Meaningful Product

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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.