The Marica‐Schönheim Inequality in Lattices

The Marica‐Schönheim Inequality says that if A is a finite family of sets, then | A −|⩾| A | where A − A =[ A 1 ∖ A 2 : A 1 , A 2 ∈ A ]. For a finite lattice L and A ⊆ L , we define a − b =∨( J a ∖ J b ) where J a =[ j ∈ L : j ⩽ a and j is join‐irreducible], and if A ⊆ L then we let A − A =[ a 1 − a 2 : a 1 , a 2 ∈ A ]. Then the analogue of the Marica‐Schöonheim Inequality is | A − A ⩾| A | for all A ⊆ L . We prove that this is true if L is distributive or complemented and modular or L is a partition lattice.