Timeline for Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets
Current License: CC BY-SA 3.0
7 events
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| Sep 3, 2015 at 9:42 | comment | added | Frank Blaga | I am trying to derive an upper bound on the sum of the size of the common prefixes of an (arbitrary) element with all incomparable elements with lower rank from an (arbitrary) antichain. This means that for choosing the greatest element and the maximal antichain this sum is actually 0, as the greatest element is comparable with all elements from the poset. I have updated the problem definition accordingly and I would be very grateful for any further feedback. | |
| Sep 3, 2015 at 9:41 | comment | added | Frank Blaga | Thank you very much for your comments @GerhardPaseman and @HughThomas! Of course, your are both correct and choosing the greatest element and the largest antichain will not only suggest and upper bound, but be the exact solution to the problem as I originally stated it. In fact, that is exactly what I did to derive my current upper bound. Your comments made me realize that I forgot to add an important additional restriction on the elements from the antichain that I need to consider: | |
| Sep 3, 2015 at 9:31 | history | edited | Frank Blaga | CC BY-SA 3.0 |
Corrected the problem definition, added further explanations
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| Sep 3, 2015 at 3:56 | comment | added | Hugh Thomas | From what you've written, it looks like you're maximizing over all $x$. Clearly, the best choice of $x$ is the all zeros vector, since it will have a length $n$ common prefix with every element of the poset. | |
| Sep 2, 2015 at 16:12 | comment | added | Gerhard Paseman | This is a good question which may be even more well received on Mathematics StackExchange. After a couple of days, if you don't get satisfaction here, I recommend posting it at math.stackexchange.com with a link to this question. Also, you might find it useful to fix n to 1, 2 , 3 successively and doing computations using those constraints. Of course, an upper bound will be suggested by letting x be an extreme element and A an extreme antichain, and the literature will support your work above. Gerhard "Post Your Computational Result Summary" Paseman, 2015.09.02 | |
| Sep 2, 2015 at 15:41 | review | First posts | |||
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| Sep 2, 2015 at 15:39 | history | asked | Frank Blaga | CC BY-SA 3.0 |