Derived semidistributive lattices

作者：Santoc****e Luigi^*

来源：Algebra Universalis, 2010, 63(2-3): 101-130.

DOI：10.1007/s00012-010-0073-4

• semidistributive lattice
• bounded lattice
• Permutohedron
• Associahedron

For L a finite lattice, let C(L) subset of L(2) denote the set of pairs gamma = (gamma(0), gamma(1)) such that gamma(0) < gamma(1) and order it as follows: gamma <= delta iff gamma(0) <= delta(0), gamma(1) not less than or equal to delta(0), and gamma(1) <= delta(1). Let C(L,gamma) denote the connected component of gamma in this poset. Our main result states that, for any gamma, C(L,gamma) is a semidistributive lattice if L is semidistributive, and that C(L,gamma) is a bounded lattice if L is bounded.
Let S(n) be the Permutohedron on n letters and let T(n) be the Associahedron on n+ 1 letters. Explicit computations show that C(S(n), alpha) = S(n-1) and C(T(n), alpha) = T(n-1), up to isomorphism, whenever alpha(1) is an atom of S(n) or T(n).
These results are consequences of new characterizations of finite join- semidistributive and of finite lower bounded lattices: (i) a finite lattice is join- semidistributive if and only if the projection sending gamma is an element of C(L) to gamma(0) is an element of 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 a generalization of the tools used by Caspard et al. to prove that lattices of finite Coxeter groups are bounded.

• 出版日期2010-5