Difference between revisions of "Invariant of an equivalence relation"

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(Created page with "{{Stub page|grade=A|msg=Find more references and flesh out}} ==Definition== Let {{M|S}} be a set and let {{M|\sim\subseteq S\times S}} be an equivalence relation on {{...")
 
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: '''Note: ''' see [[invariant]] for other uses of the term.
 
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==Definition==
 
==Definition==
−
Let {{M|S}} be a [[set]] and let {{M|\sim\subseteq S\times S}} be an [[equivalence relation]] 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}}:
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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>:
 
* 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}}.
 
** {{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}}.
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* "''{{M|\sim}} is invariant under {{M|f}}''"
 
* "''{{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)}}
 
** 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)}}
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* "''{{M|\sim}} invariance of {{M|f}}''"
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** 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.
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==Examples and instances==
 
==Examples and instances==
 
* [[Homotopy invariance of path concatenation]]
 
* [[Homotopy invariance of path concatenation]]

Latest revision as of 18:58, 9 November 2016

Note: see invariant for other uses of the term.
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Definition

Let [ilmath]S[/ilmath] be a set and let [ilmath]\sim\subseteq S\times S[/ilmath] be an equivalence relation[Note 1] on [ilmath]S[/ilmath], let [ilmath]W[/ilmath][Note 2] be any set and let [ilmath]f:S\rightarrow W[/ilmath] be any function from [ilmath]S[/ilmath] to [ilmath]W[/ilmath]. Then[1]:

  • We say "[ilmath]f[/ilmath] is an invariant of [ilmath]\sim[/ilmath]" if[Note 3]:
    • [ilmath]\forall a,b\in S[a\sim b\implies f(a)=f(b)][/ilmath] - in other words, [ilmath]f[/ilmath] is constant on the equivalence classes of [ilmath]\sim[/ilmath].

Complete invariant

With the setup of [ilmath]S[/ilmath], [ilmath]W[/ilmath], [ilmath]\sim[/ilmath] and [ilmath]f:S\rightarrow W[/ilmath] as above define a "complete invariant" as follows[1]:

  • "[ilmath]f[/ilmath] is a complete invariant of [ilmath]\sim[/ilmath]" if[Note 3]:

Terminology

It's hard to be formal in English, however we may say any of the following:

  • "[ilmath]f[/ilmath] is an invariant of [ilmath]\sim[/ilmath]"[1]
  • "[ilmath]\sim[/ilmath] is invariant under [ilmath]f[/ilmath]"
    • This makes sense as we're saying the [ilmath]a\sim b[/ilmath] property holds (doesn't vary) "under" (think "image of [ilmath]A[/ilmath] under [ilmath]f[/ilmath]"-like terminology) [ilmath]f[/ilmath], that [ilmath]f(a)=f(b)[/ilmath]
  • "[ilmath]\sim[/ilmath] invariance of [ilmath]f[/ilmath]"
    • 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

TODO: Create a category and start collecting

See also

Notes

  1. ↑ keep in mind that equality is itself an equivalence relation
  2. ↑ Think of [ilmath]W[/ilmath] as [ilmath]W\text{hatever} [/ilmath] - as usual (except in Linear Algebra where [ilmath]W[/ilmath] is quite often used for vector spaces
  3. ↑ 3.0 3.1 See "definitions and iff"

References

  1. ↑ 1.0 1.1 1.2 Advanced Linear Algebra - Steven Roman
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