Timeline for Combinatorial classes where not almost all objects are asymmetric
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 24 at 2:17 | comment | added | Richard Stanley | Interval orders and semiorders are examples. They are part of a more general setup described in EC1, second edition, Exercise 3.17. | |
| Jan 23 at 22:42 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jan 23 at 17:11 | comment | added | verret | @TimCampion Your statement about trees is not correct. Start with three paths of distinct lengths and identify one of their endpoints (so it becomes of degree three). This is a tree with trivial automorphism group. (The smallest example has order 7.) | |
| Jan 23 at 16:18 | comment | added | Tim Campion | This also reminds me of NIP theories in model theory, since a structure has trivial automorphism group iff all of its elements (and hence by finiteness all of its subsets) are definable in terms of the structure. | |
| Jan 23 at 16:13 | comment | added | Tim Campion | In fact every tree with at least two nodes admits a nontrivial automorphism. For if there are two leaves with an edge to the same node, we can swap them. Otherwise, repeat that argument with the nodes that are one edge away from being leaves. Continue in this manner, working inward from the leaves -- by connectivity of the tree we must eventually find a transposition which is an automorphism of the tree. | |
| Jan 23 at 15:27 | comment | added | Tim Campion | Finite abelian groups which are not 2-groups have a nontrivial involution given by negation. That said, given $\mathcal C$, cant you define an object of $\mathcal C'_n$ to consist of two objects of $\mathcal C_{n/2}$ and an isomorphism between them? Then $\mathcal C'$ is a combinaorial class where everything has an automorphism. Finally, I really think the language of combinatorial species seems relevant here... | |
| Jan 23 at 15:12 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jan 23 at 15:02 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jan 23 at 15:01 | comment | added | Daniel Weber | Partitions are also an example, almost all partitions have repeated parts | |
| Jan 23 at 14:58 | comment | added | Daniel Weber | A bit more generally it's that case for functions $[n] \to [n]$, and even $[n] \to [n^2]$ (birthday paradox), but I agree that it's not too interesting | |
| Jan 23 at 14:50 | comment | added | Daniel Weber | Is the set of binary strings (with isomorphisms being permutations) a valid example? For $n > 2$ no objects are asymmetric there | |
| Jan 23 at 14:31 | history | asked | Sam Hopkins | CC BY-SA 4.0 |