Definition:Concatenation of Ordered Tuples

Definition
Let $S$ be a set.

Let $w, w'$ be finite sequences in $S$ of lengths $n$ and $n'$, respectively.

Then the concatenation of $w$ and $w'$, denoted $w * w'$ or simply $w w'$, is the sequence of $n + n'$ terms defined by:


 * $w * w' \left({i}\right) := \begin{cases}

w \left({i}\right) & \text{if $1 \le i \le n$}\\ w' \left({i - n}\right) & \text{if $n < i \le n + n'$} \end{cases}$

Algebraic Structure
Let $S^*$ be the Kleene closure of $S$.

Then $\left({S^*, *}\right)$ is an algebraic structure, and by definition a magma.