Timeline for Rota's cross-cut theorem - special case?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 19, 2020 at 20:20 | comment | added | H A Helfgott | So, should I call it an immediate consequence of Rota's crosscut theorem (to which I might as well give a self-contained proof)? Or is it not really an immediate consequence? | |
| Aug 19, 2020 at 18:33 | comment | added | Timothy Chow | I normally think of Rota's crosscut theorem not as an inequality but as an equality (giving a formula for the Möbius function), so for that reason alone I would hesitate to call your lemma a "special case" of Rota's theorem. As for whether it's equivalent, Rota's theorem applies to an arbitrary geometric lattice, and unless I'm missing something, it doesn't seem that an arbitrary geometric lattice can be easily reduced to the setting of your lemma as stated. You might try emailing Bruce Sagan directly because he has thought a lot about generalizations of Rota's theorem. | |
| Aug 19, 2020 at 14:19 | comment | added | H A Helfgott | I meant the cross-cut theorem for the Möbius function. | |
| Aug 19, 2020 at 14:09 | comment | added | Benjamin Steinberg | This might depend on what you mean by the cross-cut theorem. The version for the Mobius function is a consequence of the topological cross cut theorem which says the order complex of a poset is homotopy equivalent to a certain simplicial complex built from the cross cut. The Mobius function result then follows from computing Euler characteristics. | |
| Aug 19, 2020 at 13:33 | comment | added | H A Helfgott | I should add that the connection to Rota's cross-cut theorem was pointed out to me in an answer to mathoverflow.net/questions/364743/… | |
| Aug 19, 2020 at 13:32 | history | asked | H A Helfgott | CC BY-SA 4.0 |