On 5/21/12 8:11 AM, Christian Stump wrote: >> The definition of grading is that all maximal chains of every interval >> have the same length. This is given in the above poset, but sage says >> it is not graded. > > The problem here is what people consider a "rank function" on a poset. > Do we want that all minimal elements have value 0 ?
No. I think this is not the correct definition. Then why not maximal element? It does not seem like a sensible definition. I agree that the algorithm to compute the rank function might not be so nice any longer, but that should not be a reason to use the wrong definition. > Then your poset is > indeed not graded. Or do we only want that some element has value 0 ? Yes, some element has rank 0. Or in fact all that I care about is that if y covers x then (*) rank(y) = rank(x)+1. The normalization is not that important. > Then your poset is graded. But if we take the later, what are then the > "levels" of a poset? Are all minimal elements in level 0 (which seems > to me like a good choice), or are the levels given by your grading? The grading is defined by (*). If the poset is connected, then this completely fixes the rank function up to an overall additive constant. If there are disconnected components, one may want to use a particular convention for the additive constant. > I think you're right since the existence of a rank function seems to > be common sense as the definition > (http://en.wikipedia.org/wiki/Graded_poset) (this is equivalent to > your definition for locally finite posets). But then, we have to make > a choice for the levels of a poset. > > My 2 cents, Christian Best, Anne -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To post to this group, send email to sage-combinat-devel@googlegroups.com. To unsubscribe from this group, send email to sage-combinat-devel+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.
