The number of meets between two subsets of a lattice

Abstract

Let L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove | A| | B| ⩽ | L| | A ∧ B| for any A, B ⊃ L, where |·| denotes cardinality and A ∧ B = { a ∧ b: a ϵ A, b ϵ B}. | A ∧ B| attains its minimum for fixed | A|, | B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed | A|, | B|, | A ∧ B| is minimized when A and B are the | A| and | B| smallest n-digit k-ary numbers written backwards and forwards, respectively. | A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made.