Timeline for What classes of groups can arise as "symmetry groups of terms"?
Current License: CC BY-SA 4.0
10 events
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| Jan 30, 2022 at 6:16 | comment | added | Tim Campion | Note that if you allow your structure to have two sorts, you can modify the example to have $\{a,b\}$ be the carrier for one sort and $\{1,2,\dots\}$ be the carrier for a second sort. Then I think you get that $G(\mathfrak A)$ in Noah's sense is precisely $\mathcal K$ (where all of the $f_G$'s go from the second sort to the first) | |
| Dec 30, 2021 at 21:18 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Dec 30, 2021 at 9:38 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Dec 30, 2021 at 6:53 | comment | added | Peter LeFanu Lumsdaine | Just to clarify: this answer’s argument “the variable-appearance condition doesn’t restrict” shows that “for any algebra $A$, there’s an expanded algebra $A'$ such that the symmetry-groups of Noah-restricted terms of $A'$ are the symmetry groups of arbitrary terms of $A$” . But it doesn’t imply the converse, as Noah’s first comment here shows. | |
| Dec 27, 2021 at 4:15 | comment | added | Noah Schweber | I've added a more focused subquestion to hopefully clarify the sort of thing I'm looking for. | |
| Dec 26, 2021 at 23:58 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Dec 26, 2021 at 23:51 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Dec 26, 2021 at 23:28 | comment | added | Noah Schweber | This is interesting but I don't think it answers my question. Shifting to terms which depend on all variables changes the situation quite a bit: for the $\mathbb{G}(\mathfrak{A})$s as defined in my question, having a single nontrivial group automatically forces us to have groups of arbitrarily large finite order. And dropping the variable-appearance restriction **does** change the question: if we don't require every variable to actually appear in each term, then $\mathbb{G}(\mathfrak{A})$ must be closed under taking direct products with $S_n$s via dummy variables. | |
| Dec 26, 2021 at 23:10 | history | edited | Keith Kearnes | CC BY-SA 4.0 |
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| Dec 26, 2021 at 22:54 | history | answered | Keith Kearnes | CC BY-SA 4.0 |