Derived semidistributive lattices

Abstract

For L a finite lattice, let C(L) ⊆ L 2 denote the set of pairs γ = (γ 0 , γ 1 ) such that γ 0 ≺ γ 1 and order it as follows: γ ≤ δ iff γ 0 ≤ δ 0 , γ 1 ≤ δ 0 , and γ 1 ≤ δ 1 . Let C(L, γ) denote the connected component of γ in this poset. Our main result states that, for any γ, C(L, γ) is a semidistributive lattice if L is semidistributive, and thatLet S n be the Permutohedron on n letters and let Tn be the Associahedron on n+1 letters. Explicit computations show that C(Sn, α) = S n−1 and C(Tn, α) = T n−1 , up to isomorphism, whenever α 1 is an atom of Sn or Tn.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 γ ∈ C(L) to γ 0 ∈ 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.

The Canonical Join Complex

“…The discussion in Section 4 does not constitute a complete list of lattice theoretic operations which preserve (join)-semidistributivity. For example, the derived lattice CpLq discussed in [29], the box product defined in [17] (see also, [35,Corollary 8.2]), and the lattice of multichains from [20] all preserve (join)-semidistributivity.…”