Excess Problem for Modular Lattices

Abstract

In this paper, we show that for every modular lattice \(L\), if its size is at least three times its excess, then each component of its direct product decomposition is isomorphic to one of the following: a Boolean lattice of rank one \(B_1\), a chain of length two \(3\), a diamond \(M_3\), and \(M_4\), where \(M_n\) is a modular lattice of rank two which has exactly \(n\) atoms.