Timeline for How to describe this set of maps of posets?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2022 at 7:46 | vote | accept | Philippe Gaucher | ||
| Sep 7, 2022 at 7:38 | answer | added | Philippe Gaucher | timeline score: 1 | |
| Jul 29, 2022 at 9:06 | comment | added | Philippe Gaucher | @PeterTaylor In the language of directed homotopy, the associated map from $[0,1]^n$ to itself not only will take a directed path to a directed path, but also the $L_1$-arc length from $0^n$ will be preserved (here it coincides with the distance for the $d_1$ metric): different words for the same phenomenon. | |
| Jul 29, 2022 at 8:54 | comment | added | Peter Taylor | The Hamming distance is the number of positions in which the tuple/word differs. (The (Hamming) weight is the Hamming distance from $0^n$, and I think that your comment is intended to say that the Hamming weight of $f(w)$ is the Hamming weight of $w$, which follows easily from the property that the map is strictly increasing). | |
| Jul 29, 2022 at 8:47 | comment | added | Philippe Gaucher | @PeterTaylor I don't know what the Hamming distance is but all such $f$ have the property that $\epsilon_1+\dots+\epsilon_n=f(\epsilon_1)+\dots+f(\epsilon_n)$. | |
| Jul 29, 2022 at 8:26 | comment | added | Peter Taylor | More generally, if two words $w_1, w_2$ of weight $k$ have Hamming distance 2, the Hamming distance of $f(w_1), f(w_2)$ must be at most 2, and if it is equal to 2 then $f(w_1 \vee w_2) = f(w_1) \vee f(w_2)$. | |
| Jul 29, 2022 at 8:22 | comment | added | Peter Taylor | @SamHopkins, some of them are easy to eliminate. In particular, if the second-lowest rank has all $n$ tuples with total weight 1 then the map must be bijective. To take another example, if $n=3$ and the ranks have restrictively 1,2,2,1 elements, one of the elements at rank value 1 has two successors at rank level 2, and the other has only one. | |
| Jul 28, 2022 at 15:30 | comment | added | Sam Hopkins | Even classifying the images of such maps seems highly nontrivial. I don’t see why you couldn’t get e.g. every rank n graded poset this way. | |
| Jul 28, 2022 at 8:30 | history | asked | Philippe Gaucher | CC BY-SA 4.0 |