View source for Invariant of an equivalence relation

: '''Note: ''' see [[invariant]] for other uses of the term. 
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==Definition==
Let {{M|S}} be a [[set]] and let {{M|\sim\subseteq S\times S}} be an [[equivalence relation]]<ref group="Note">keep in mind that [[equality]] is itself an equivalence relation</ref> on {{M|S}}, let {{M|W}}<ref group="Note">Think of {{M|W}} as {{M|W\text{hatever} }} - as usual (except in [[Linear Algebra (subject)|Linear Algebra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. Then{{rALASR}}:
* We say "{{M|f}}'' is an invariant of ''{{M|\sim}}" if<ref group="Note" name="definition">See "''[[definitions and iff]]''"</ref>:
** {{M|1=\forall a,b\in S[a\sim b\implies f(a)=f(b)]}} - in other words, {{M|f}} is [[constant on]] the [[equivalence classes]] of {{M|\sim}}.
===Complete invariant===
With the setup of {{M|S}}, {{M|W}}, {{M|\sim}} and {{M|f:S\rightarrow W}} as above define a "''complete invariant''" as follows<ref name="ALASR"/>:
* "{{M|f}}'' is a complete invariant of {{M|\sim}}''" if<ref group="Note" name="definition"/>:
** {{M|1=\forall a,b\in S[a\sim b\iff f(a)=f(b)]}} - in other words, {{M|f}} is [[constant on]] ''and'' [[distinct on]] the [[equivalence classes]] of {{M|\sim}}.
==Terminology==
It's hard to be formal in English, however we may say any of the following:
* "''{{M|f}} is an invariant of {{M|\sim}}''"<ref name="ALASR"/>
* "''{{M|\sim}} is invariant under {{M|f}}''"
** This makes sense as we're saying the {{M|a\sim b}} property holds (doesn't vary) "under" (think "image of {{M|A}} under {{M|f}}"-like terminology) {{M|f}}, that {{M|1=f(a)=f(b)}}
* "''{{M|\sim}} invariance of {{M|f}}''" 
** This works better when the relations have names, eg "''equality invariance of Alec's heuristic''" (that's a made up example) and this would be a proposition or a claim.

==Examples and instances==
* [[Homotopy invariance of path concatenation]]
** [[Homotopy invariance of loop concatenation]]
{{Todo|Create a category and start collecting}}
==See also==
* [[Complete system of invariants]] - a finite set of complete invariants really.
* [[Set of canonical forms]] - a [[subset of]] {{M|S}}, {{M|C\in\mathcal{P}(S)}}, such that there [[exists a unique]] {{M|c\in C}} such that {{M|c\sim s}}
** An [[equivalent conditions to the axiom of choice|equivalent condition to]] the [[axiom of choice]] is that [[every partition has a set of representatives]] that's closely related. Be warned

==Notes==
<references group="Note"/>
==References==
<references/>
{{Requires references|grade=A|msg=More on the fundamentals of mathematics would be good}}
{{Definition|Set Theory|Abstract Algebra|Elementary Set Theory}}