Invariant of an equivalence relation
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- 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]}:
- [ilmath]\forall a,b\in S[a\sim b\iff f(a)=f(b)][/ilmath] - in other words, [ilmath]f[/ilmath] is constant on and distinct on the equivalence classes of [ilmath]\sim[/ilmath].
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
- Complete system of invariants - a finite set of complete invariants really.
- Set of canonical forms - a subset of [ilmath]S[/ilmath], [ilmath]C\in\mathcal{P}(S)[/ilmath], such that there exists a unique [ilmath]c\in C[/ilmath] such that [ilmath]c\sim s[/ilmath]
- An equivalent condition to the axiom of choice is that every partition has a set of representatives that's closely related. Be warned
Notes
- ↑ keep in mind that equality is itself an equivalence relation
- ↑ 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
- ↑ ^{3.0} ^{3.1} See "definitions and iff"
References
- ↑ ^{1.0} ^{1.1} ^{1.2} Advanced Linear Algebra - Steven Roman
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More on the fundamentals of mathematics would be good