Timeline for Interesting uniform posets
Current License: CC BY-SA 4.0
12 events
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| Mar 16, 2022 at 12:55 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Mar 14, 2022 at 19:22 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Mar 13, 2022 at 22:28 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Mar 13, 2022 at 20:04 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Mar 4, 2022 at 3:16 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Mar 2, 2022 at 2:20 | comment | added | Richard Stanley | Example 3.18.3 in EC1 leads to lots of uniform sequences of posets. In fact, they have the stronger property that all intervals of $P_n$ of length $m$ are isomorphic to $P_m$. Not all binomial posets have the property that all intervals of length $n$ are isomorphic, but those of Example 3.18.3 have this property if in (d) we take $P_1,\dots,P_k$ to have this property, and similarly for $P$ in (f). | |
| Mar 1, 2022 at 23:31 | comment | added | Joseph Van Name | The Krull-Schmidt theorem also applies to modules. If $M$ is a finite indecomposable module, and $P_{n}$ is the collection of all submodules of $M^{n}$ that are direct factors of $M^{n}$, then $P_{n}$ is uniform since each element of $P^{n}$ is of the form $M^{m}$ for some $m\leq n$. One can produce more examples of uniform posets by letting $P_{n}$ denote the collection of all cosets of direct factors of $M^{n}$ (or of $G^{ n}$ in the group case). | |
| Mar 1, 2022 at 23:24 | comment | added | Joseph Van Name | Let $G$ be a finite directly indecomposable group (directly indecomposable means that the group cannot be written as a direct product of subgroups). Let $P_{n}$ be the collection of all subgroups of $G^{n}$ that are internal direct factors of $G$. Then as a consequence of the Krull-Schmidt theorem, all groups in $P_{n}$ are isomorphic to the group $G^{m}$ for some $0\leq m\leq n$, so the posets $(P_{n})_{n}$ are uniform. | |
| Mar 1, 2022 at 20:47 | comment | added | Joseph Van Name | By the affine subspaces, I mean the collection of all cosets of the vector subspaces of $\mathbb{F}_{q}^{i}$. The minimal object is $\emptyset$ while the maximal object is $\mathbb{F}_{q}^{i}$ itself. In the case where $q=p$ (so $q$ is prime), the affine subspaces are simply the subsets which are closed under the operation $(x,y,z)\mapsto x-y+z$. This operation is the heap operation on $F_{p}^{i}$ en.wikipedia.org/wiki/Heap_%28mathematics%29. | |
| Mar 1, 2022 at 20:09 | comment | added | Sam Hopkins | @JosephVanName: I don't understand the order you have in mind (in particular, why is there always a minimum and maximum element for that poset)? | |
| Mar 1, 2022 at 20:06 | comment | added | Joseph Van Name | The lattice of affine subspaces of $\mathbb{F}_{q}^{i}$ is another example. | |
| Mar 1, 2022 at 17:23 | history | asked | Sam Hopkins | CC BY-SA 4.0 |