Frankl's conjecture is true for modular lattices
Frankl's Conjecture states that every finite lattice \(L\) contains a join-irreducible element \(x\) generating a principal filter \(V_x\) of at most half the size of \(L\). The authors observe that \(| V_x|\leq{1\over 2}| L|\) holds for all \(x\) such that for all \(y>x\) there is \(z< y\) such that that \(z\vee x= y\). And they prove that a minimal such \(x\) is join-irreducible if \(L\) is modular. Thus, they establish Frankl's Conjecture for modular lattices. Moreover, for the case of modular lattices as well as the case of sectionally complemented lattices, they show that there is an irreducible \(x\) such that \(| V_x|={1\over 2}| L|\) if and only if \(L\) is Boolean. This paper makes a substantial contribution, but, as the authors remark, there are lattices with no irreducibles \(x\) of the above type.
scientific article; zbMATH DE number 1234335