\begin{definition}[Definition:Concatenation of Ordered Tuples]
Let $S$ be a set.
Let $w, w'$ be ordered tuples in $S$ of lengths $n$ and $n'$, respectively.
Then the '''concatenation of $w$ and $w'$''', denoted $w * w'$ or simply $w w'$, is the ordered tuple of $n + n'$ terms defined by:
:$\map {w * w'} i := \begin{cases}
\map w i & : \text {if $1 \le i \le n$} \\
\map {w'} {i - n} & : \text {if $n < i \le n + n'$}
\end{cases}$
\end{definition}