# Definition:Ordered Tuple/Defined by Sequence

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of advanced complexity.In particular: Especially when the third definition is included, it needs to be considered to put this (technical, specialised) definition separate from the main Ordered Tuple articleUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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 $A$.

Work In ProgressIn particular: page proving existence and uniqueness of isomorphismYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

## Sources

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