Derived Semidistributive Lattices

Let C(L) denote the set of covers of a poset L: γ ∈ C(L) if and only γ = (γ0, γ1) ∈ L×L and the interval {x | γ0 ≤ x ≤ γ1 } is a two elements poset. If L is a lattice then there is a natural ordering of C(L): γ ≤ δ if and only if γ0 ≤ δ0, γ1 6≤ δ0, and γ1 ≤ δ1. That is, γ ≤ δ if and only if the cover γ transposes up to δ. For α ∈ C(L) let C(L,α) denote the component of the poset C(L) connected to α. For example, if L is finite join semidistributive and α = (j∗, j) for a join irreducible j and its unique lower cover j∗, then C(L,α) = {β |α ≤ β }. The main result we wish to present is the following: