Timeline for Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?
Current License: CC BY-SA 4.0
11 events
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| Feb 8 at 12:56 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Feb 25, 2020 at 13:37 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Feb 19, 2020 at 15:43 | comment | added | Will Brian | I take back what I said about potential isomorphisms. (But I don't take back what I said about your question being a good question.) | |
| Feb 19, 2020 at 14:02 | comment | added | YCor | Thanks, I eventually posted the question mathoverflow.net/questions/353074 | |
| Feb 19, 2020 at 13:39 | comment | added | Will Brian | That's a good question. I don't think it's known. My guess is that the answer is yes, and we might be able to prove they are elementarily equivalent by showing that they're potentially isomorphic (en.wikipedia.org/wiki/Potential_isomorphism). I don't think that the set of all homeomorphisms satisfying some fixed sentence needs to be Borel -- this isn't true for sets of reals, anyway, because unbounded quantifiers push us out of the Borel sets into the projective hierarchy. | |
| Feb 19, 2020 at 13:28 | comment | added | YCor | Do you know if $(\mathrm{Clopen}(\beta^*\omega),\Phi)$ and $(\mathrm{Clopen}(\beta^*\omega),\Phi^{-1})$ are elementary equivalent (as structures encoding a Boolean algebra endowed with an automorphism)? I might post a separate question if it's not immediate or an immediate consequence of your result (it's not clear to me that the set of homeomorphisms $\alpha$ satisfying some sentence $u(\alpha)$ is Borel). | |
| Feb 18, 2020 at 16:50 | comment | added | Will Brian | You're welcome! This is a question that I've thought a lot about, so I'm happy to be able to share. | |
| Feb 18, 2020 at 16:49 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Feb 18, 2020 at 16:35 | comment | added | YCor | Thanks for this detailed answer and for the links! (Side note: I had to open the second link to figure out that "isomorphism class" denotes what I know as "conjugacy class" — I agree that from a certain categorical viewpoint it's also an isomorphism class, namely in the category of spaces endowed with a homeo; actually my colleagues in dynamical systems talk of conjugacy of dynamical systems even when acting on different spaces) | |
| Feb 18, 2020 at 16:27 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Feb 18, 2020 at 16:20 | history | answered | Will Brian | CC BY-SA 4.0 |