Timeline for Constructing a smooth lattice from a discrete one.
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2010 at 23:39 | comment | added | Gerhard Paseman | Like Dedekind completed the linear order of the rationals using downsets to make reals, so one can do so for lattices to turn a lattice into a complete lattice. This is Dedekind-MacNeille completion. Also, see if you can provide satisfactory examples on M3, N5, and lattices which are incidence relations of small projective spaces. If you know what your result looks like for those examples, then maybe an appropriate generalization can be found for other finite lattices. (E.g., for the Fano plane, draw 7 line dots above 7 point dotss, each line covers the three points it contains.) | |
| Feb 4, 2010 at 17:37 | comment | added | Mariano Suárez-Álvarez | @Suresh: If you want the interior, and so on, you maybe want something like the order complex of the poset, or the geometrical realization. I do not think you can turn these into posets in any significant way, though. | |
| Feb 4, 2010 at 8:53 | comment | added | Suresh Venkat | Thanks ! What is a Dedekind/MacNeille completion ? In response to your suggestion, replacing the line by a copy of [0,1] makes sense, but this only gives me a "skeleton". For example, if I have a diamond in the Hasse diagram consisting of $a$ below b and c, which are below d, I'd like to access the "interior" of this diamond as well. | |
| Feb 4, 2010 at 8:46 | history | answered | Gerhard Paseman | CC BY-SA 2.5 |