Derived Semidistributive Lattices
| dc.creator | Santocanale, Luigi | |
| dc.date | 2007-08-13 | |
| dc.date | 2008-07-21 | |
| dc.date.accessioned | 2026-07-07T09:51:36Z | |
| dc.date.available | 2026-07-07T09:51:36Z | |
| dc.description | For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in this poset. Our main result states that C(L,g) is a semidistributive lattice if L is semidistributive, and that C(L,g) is a bounded lattice if L is bounded. Let S_n be the permutohedron on n letters and T_n be the associahedron on n+1 letters. Explicit computations show that C(S_n,a) = S_{n-1} and C(T_n,a) = T_{n-1}, up to isomorphism, whenever a is an atom. These results are consequences of new characterizations of finite join semidistributive and finite lower bounded lattices: (i) a finite lattice is join semidistributive if and only if the projection sending g in C(L) to g_0 in L creates pullbacks, (ii) a finite join semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are generalization of the tools used by Barbut et al. to prove that lattices of Coxeter groups are bounded. | |
| dc.identifier | https://arxiv.org/abs/0708.1695 | |
| dc.identifier | http://arxiv.org/abs/0708.1695 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/165305 | |
| dc.subject | Logic | |
| dc.subject | Combinatorics | |
| dc.subject | Category Theory | |
| dc.subject | Rings and Algebras | |
| dc.subject | 06B05, 06A07, 18B35 | |
| dc.title | Derived Semidistributive Lattices | |
| dc.type | text |