Derived semidistributive lattices

For L a finite lattice, let ${\mathbb {C}(L) \subseteq L^2}$ denote the set of pairs γ = (γ ₀, γ ₁) such that ${\gamma_0 \prec \gamma_1}$ and order it as followsγ ≤ δ iff γ ₀ ≤ δ ₀, ${\gamma_{1} \nleq \delta_0,}$ and γ ₁ ≤ δ ₁. Let ${\mathbb {C}(L, \gamma)}$ denote the connected component of γ in this poset. Our main result states that, for any ${\gamma, \mathbb {C}(L, \gamma)}$ is a semidistributive lattice if L is semidistributive, and that ${\mathbb {C}(L, \gamma)}$ is a bounded lattice if L is bounded. Let ${\mathcal{S}_{n}}$ be the Permutohedron on n letters and let ${\mathcal{T}_{n}}$ be the Associahedron on n + 1 letters. Explicit computations show that ${\mathbb {C}(\mathcal{S}_{n}, \alpha) = \mathcal{S}_{n-1}}$ and ${\mathbb {C}(\mathcal {T}_n, \alpha) = \mathcal {T}_{n-1}}$ , up to isomorphism, whenever α1 is an atom of ${\mathcal{S}_{n}}$ or ${\mathcal{T}_{n}}$ . 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 ${\gamma \in \mathbb {C}(L)}$ to ${\gamma_0 \in L}$ 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 a generalization of the tools used by Caspard et al. to prove that lattices of finite Coxeter groups are bounded.