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:


 * $\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}$

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

Then $\struct {S^*, *}$ is an algebraic structure, and by definition a magma.