Timeline for What classes of groups can arise as "symmetry groups of terms"?
Current License: CC BY-SA 4.0
29 events
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| S Feb 6, 2022 at 23:07 | history | bounty ended | CommunityBot | ||
| S Feb 6, 2022 at 23:07 | history | notice removed | CommunityBot | ||
| Jan 30, 2022 at 5:50 | comment | added | Noah Schweber | Put another way, I'm interested in the (hyper)class $\{\mathbb{G}(\mathfrak{A}): \mathfrak{A}$ is an algebra$\}$, and this class is richer when we do require variables to appear. | |
| Jan 30, 2022 at 5:48 | comment | added | Noah Schweber | @TimCampion Yes, $y$ appears in that term (and terms representing equal functions aren't equal), but that's a different algebra $\mathfrak{A}'$ with a different associated collection of groups $\mathbb{G}(\mathfrak{A}')$. To see that requiring variables to appear actually makes a difference, note that if we dropped that requirement we would always have every $\mathbb{G}(\mathfrak{A})$ be closed under direct products with $S_n$s, which isn't the case when we do require variables to appear. Basically, requiring variables to appear gives more possibilities for the $\mathbb{G}(\mathfrak{A})$s. | |
| Jan 30, 2022 at 5:45 | comment | added | Tim Campion | If $\mathfrak A$ is an algebra, I can enlarge the language of $\mathfrak A$ by a binary function $f$ with the equation $f(x,y) = x$. If I take my term to be $t(x,y) = f(x,y)$, then is $y$ considered to "appear" in this term? If so, then I think the "variable appearance" condition is not very strong (when we have the flexibility to choose our language to include things like $f$). If not, then I don't think I understand the "variable appearance" condition. Basically -- are terms considered equal if they are represent the same function in $\mathfrak A$? | |
| Jan 30, 2022 at 5:37 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jan 30, 2022 at 5:35 | comment | added | Noah Schweber | @TimCampion Yes, but "algebra" is shorter and standardly used in this sort of context. | |
| Jan 30, 2022 at 5:34 | comment | added | Tim Campion | Ah, I see. So you could equally have said "structure in a functional language" everywhere rather than "algebra". | |
| Jan 30, 2022 at 5:32 | comment | added | Noah Schweber | @TimCampion I don't talk about varieties anywhere; everything is about individual algebras, not collections of algebras. As to why I'm looking at algebras as opposed to arbitrary first-order structures, they just seem particularly interesting. | |
| Jan 30, 2022 at 5:30 | comment | added | Tim Campion | What's the significance of restricting to varieties, rather than (say) arbitrary first-order theories? Is it meant to make the problem more tractable? | |
| Jan 30, 2022 at 5:20 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jan 30, 2022 at 5:12 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Jan 30, 2022 at 0:53 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| S Jan 29, 2022 at 22:06 | history | bounty started | Noah Schweber | ||
| S Jan 29, 2022 at 22:06 | history | notice added | Noah Schweber | Draw attention | |
| Jan 29, 2022 at 22:06 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| S Jan 6, 2022 at 23:04 | history | bounty ended | CommunityBot | ||
| S Jan 6, 2022 at 23:04 | history | notice removed | CommunityBot | ||
| Dec 30, 2021 at 6:09 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| S Dec 29, 2021 at 21:23 | history | bounty started | Noah Schweber | ||
| S Dec 29, 2021 at 21:23 | history | notice added | Noah Schweber | Draw attention | |
| Dec 27, 2021 at 4:26 | history | became hot network question | |||
| Dec 27, 2021 at 4:15 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 27, 2021 at 2:24 | comment | added | Noah Schweber | @YCor I would also be interested in that version of the question, but right now at least I'm primarily thinking about the groups on their own. (And besides, a coarser invariant is also easier to fully analyze.) | |
| Dec 27, 2021 at 0:54 | comment | added | YCor | Why asking about isomorphism type as groups, rather than isomorphism type as permutation groups (which sounds more meaningful — I guess this is what's meant by "isomorphism types of subgroups of symmetric group" in Keith Kearnes' answer? | |
| Dec 26, 2021 at 23:25 | history | rollback | Noah Schweber |
Rollback to Revision 1
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| Dec 26, 2021 at 23:23 | history | edited | Noah Schweber | CC BY-SA 4.0 |
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| Dec 26, 2021 at 22:54 | answer | added | Keith Kearnes | timeline score: 13 | |
| Dec 26, 2021 at 20:19 | history | asked | Noah Schweber | CC BY-SA 4.0 |