Timeline for Classification results
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2022 at 4:39 | comment | added | Wlod AA | I've corrected the OT's "Theorem". | |
| Oct 21, 2022 at 4:36 | history | edited | Wlod AA | CC BY-SA 4.0 |
The correct formulation!
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| Oct 21, 2022 at 2:27 | answer | added | Timothy Chow | timeline score: 6 | |
| Oct 20, 2022 at 14:29 | comment | added | Ben McKay | @user493267: true: If you set $I(X)=X$, then the second theorem becomes trivial. What I am saying is that if you have a list containing one element in each isomorphism equivalence class, as is supposed in the first theorem, then you can map the entire class $C$ to the elements of that list by a map taking each element to the unique isomorphic element of the list, defining a map $I$ which makes the second theorem true. But this is only interesting if we expect, in circumstances likely to be encountered, that we could decide which element of the list a given element $X$ of $C$ is isomorphic to. | |
| Oct 20, 2022 at 12:01 | comment | added | user493267 | @YCor "A classification should include both results" But what is the relationship between these two results?? | |
| Oct 20, 2022 at 12:00 | comment | added | user493267 | @BenMcKay "A classification theorem of the first type tautologically gives one of the second, where 𝐼(𝑋) is the element 𝑌 of the list which is isomorphic to 𝑋." If you set $I(X):=X$, then the second theorem becomes trivial - even without using the first theorem!! So I don't get what you are saying. | |
| Oct 20, 2022 at 10:01 | comment | added | Francesco Polizzi | Having a complete invariant which is also effectively computable is quite a rare thing. | |
| Oct 20, 2022 at 9:04 | comment | added | Ben McKay | A classification theorem of the first type tautologically gives one of the second, where $I(X)$ is the element $Y$ of the list which is isomorphic to $X$. Of course, the real issue is how the object $X$ is presented, so that we can see what sort of maps $I$ are computable in terms of that presentation. | |
| Oct 20, 2022 at 8:49 | comment | added | YCor | A classification should include both results, and it is indeed the case in the example of finite simple groups, and closed surfaces. Just it turns out that often the first part is by far the hardest, notably in the case of the classification of finite simple groups. For instance, for the classification of 7-dimensional nilpotent Lie complex algebras, I think that several initially publish lists were slightly flawed because they failed to correctly characterize isomorphism. | |
| Oct 20, 2022 at 8:48 | history | edited | YCor |
edited tags
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| S Oct 20, 2022 at 7:41 | review | First questions | |||
| Oct 20, 2022 at 9:02 | |||||
| S Oct 20, 2022 at 7:41 | history | asked | user493267 | CC BY-SA 4.0 |