Two and a Half Remarks on the Marica‐Schönheim Inequality

The Marica‐Schönheim inequality states that the number of distinct differences of the form A\B, with A, B taken from a given finite family A of sets is at least | A |. We prove that equality occurs essentially if and only if A is the product of an ideal and a filter. We also prove an infinite version of the theorem, conjectured (in weaker form) by Daykin and Lovasz. Finally, we note that a generalization (due to Ahlswede and Daykin) of the inequality which considers two families A and B holds under a weaker assumption on the relation between A and B .