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.