Difference between revisions of "Notes:Quotient"
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Let {{M|X}} be a set and let {{M|\sim}} be an [[equivalence relation]] on the elements of {{M|X}}. | Let {{M|X}} be a set and let {{M|\sim}} be an [[equivalence relation]] on the elements of {{M|X}}. | ||
* Then {{M|\frac{X}{\sim} }} denotes the "[[equivalence class|equivalence classes]]" of {{M|~}} | * Then {{M|\frac{X}{\sim} }} denotes the "[[equivalence class|equivalence classes]]" of {{M|~}} | ||
Revision as of 16:04, 18 November 2015
Terminology
Let [ilmath]X[/ilmath] be a set and let [ilmath]\sim[/ilmath] be an equivalence relation on the elements of [ilmath]X[/ilmath].
- Then [ilmath]\frac{X}{\sim} [/ilmath] denotes the "equivalence classes" of [ilmath]~[/ilmath]
This is best thought of as a map:
- [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] by [ilmath]\pi:x\mapsto [x][/ilmath] where recall:
- [ilmath][a]=\{x\in X\vert x\sim a\}[/ilmath], the notation [ilmath][a][/ilmath] makes sense, as by the reflexive property of [ilmath]\sim[/ilmath] we have [ilmath]a\in[a][/ilmath]
Quotient structure
Suppose that [ilmath]\odot:X\times X\rightarrow X[/ilmath] is any map, and writing [ilmath]x\odot y:=\odot(x,y)[/ilmath] when does [ilmath]\odot[/ilmath] induce an 'equivalent' mapping on [ilmath]\frac{X}{\sim} [/ilmath]?
- This is a mapping: [ilmath]\odot:\frac{X}{\sim}\times\frac{X}{\sim}\rightarrow\frac{X}{\sim} [/ilmath] where [ilmath][x]\odot[y]=[x\odot y][/ilmath]
- should such an operation be 'well defined' (which means it doesn't matter what representatives we pick of [ilmath][x][/ilmath] and [ilmath][y][/ilmath] in the computation)
Alternatively
We have no concept of [ilmath]\odot[/ilmath] on [ilmath]\frac{X}{\sim} [/ilmath], but we do on [ilmath]X[/ilmath]. The idea is that:
- Given a [ilmath][x][/ilmath] and a [ilmath][y][/ilmath] we go back
- To an [ilmath]x[/ilmath] and a [ilmath]y[/ilmath] representing those classes.
- Compute [ilmath]x\odot y[/ilmath]
- Then go forward again to [ilmath][x\odot y][/ilmath]
In functional terms we may say:
- [ilmath]\odot:\frac{X}{\sim}\times\frac{X}{\sim}\rightarrow\frac{X}{\sim} [/ilmath] given by:
- [ilmath]\odot([x],[y])\mapsto\pi(\underbrace{\pi^{-1}([x])\odot\pi^{-1}([y])}_{\text{if }\odot\text{ makes sense} })=\Big[\pi^{-1}([x])\odot\pi^{-1}([y])\Big][/ilmath]
Here [ilmath]\pi^{-1}([x])[/ilmath] is a subset of [ilmath]X[/ilmath] containing exactly those things which are equivalent to [ilmath]x[/ilmath] (as these things all map to [ilmath][x][/ilmath]).
- We can say [ilmath]A\odot B[/ilmath] (for [ilmath]A\subseteq X[/ilmath] and [ilmath]B\subseteq X[/ilmath]) if [ilmath]a\odot b\sim a'\odot b'[/ilmath]
As then
- We can define [ilmath]\pi(A)[/ilmath] (for [ilmath]A\subseteq X [/ilmath]) properly if
This all seems very contrived
As a diagram
I seem to be asking when a map (dashed line) is induced such that the following diagram commutes:
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[math]\begin{xy}\xymatrix{X\times X \ar@{->}[r]^-\odot \ar@{->}[d]_-{\pi\times\pi}& X \ar@{->}[d]^-{\pi} \\ \frac{X}{\sim}\times\frac{X}{\sim} \ar@{.>}[r]_-{\odot} & \frac{X}{\sim} }\end{xy}[/math] |
| Diagram |
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