An inequality for the weights of two families of sets, their unions and intersections

Abstract

then ~(A) fi(B) < 7(A v B) cS(A A B) for all A, B ~ S, (2) where e(A) = ~(a~A) e(a) and A v B = {awb; aeA, b~B} and A A B = {ac~b; a~A, b~B}. Since every distributive lattice can be embedded in the subsets of some set we get an immediate Corollary. If S is a distributive lattice and (2) holds whenever A, B each contain exactly one point of S then (2) always holds. Here S, A, B may be infinite. Our theorem contains as special cases results of Anderson, Daykin, Fortuin, Ginibre, Greene, Holley, Kasteleyn, Kleitman, Seymour, West and others 1. We discovered it whilst guests at the Mathematisches Forschungsinstit ut Oberwolfach and thank all concerned for their kindness to us.