# Definition:Concatenation of Ordered Tuples

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*This page is about concatenation of finite sequences. For other uses, see Definition:Concatenation.*

## 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.

## Linguistic Note

The word **concatenation** derives from the Latin word **catena** for **chain**.

However, the end result of such an operation is not to be confused with a (set theoretical) chain.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.7$