Talk:Invariant (mathematics)

	This  level-5 vital article is rated C-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:
Mathematics High‑priority Mathematics portal This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. High This article has been rated as High-priority on the project's priority scale.

Text and/or other creative content from this version of Invariant (computer science) was copied or moved into Invariant (mathematics) with this edit on 16:04, 25 October 2018. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists.

Archives

1

This page has archives. Sections older than 365 days may be auto-archived by Lowercase sigmabot III if there are more than 3.

Invariant or closed

I think that the definition of an invariant set is not correct. If I am not mistaken, a set ${\displaystyle S}$ is invariant under a map ${\displaystyle T}$ if ${\displaystyle a\in S\Leftrightarrow T(a)\in S}$. In particular, if ${\displaystyle T}$ is a bijection (which is the typical case where this terminology is used), this means that ${\displaystyle S=T(S)}$, i.e. ${\displaystyle S}$ is a fixed element of the action of ${\displaystyle T}$ in the power set. I think the current definition corresponds to "closed": a set ${\displaystyle S}$ is closed under ${\displaystyle T}$ if ${\displaystyle T(S)\subseteq S}$. Another common terminology is "fixed". A set ${\displaystyle S}$ is (pointwise) fixed by ${\displaystyle T}$ if ${\displaystyle T(a)=a}$ for all ${\displaystyle a\in S}$, and setwise fixed if ${\displaystyle T(S)=S}$. So setwise fixed is a synonym for invariant (if ${\displaystyle T}$ is a bijection). However, these terms are sometimes confused: a good example is pointwise invariant and setwise invariant instead of pointwise fixed and setwise fixed. As mentioned above, another common term is "stable", which I believe is also a synonym for "invariant", and not for "closed". In any case, it might be good to find references to the different uses. arf