# Definition:Ordered Tuple/Defined by Sequence

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## Definition

Let $\sequence {a_k}_{k \mathop \in A}$ be a finite sequence of $n$ terms.

Let $\sigma$ be a permutation of $A$.

Then the **ordered $n$-tuple defined by the sequence $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$** is the ordered $n$-tuple:

- $\sequence {a_{\map \sigma {\map \tau j} } }_{1 \mathop \le j \mathop \le n}$

where $\tau$ is the unique isomorphism from the totally ordered set $\closedint 1 n$ onto the totally ordered set $A$.

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## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations