The Marica-Schönheim Inequality in Lattices
Zsolt Lengvárszky · Bulletin of the London Mathematical Society · 1996
The Marica-Schönheim Inequality says that if A is a finite family of sets, then |A−|⩾|A| where A−A=[A1∖A2:A1, A2∈A]. For a finite lattice L and A⊆L, we define a−b=∨(Ja∖Jb) where Ja=[j∈L:j⩽a and j is join-irreducible], and if A⊆L then we let A−A=[a1−a2: a1, a2∈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.