CHAPTER 4 - MODULAR LATTICES

Abstract

This chapter discusses modular lattices. In order theory, a modular lattice is a lattice that satisfies the self-dual condition. A sublattice of a modular lattice is modular. However, a lattice is modular only if it does not contain a sublattice isomorphic with the pentagonal lattice. In the case of lattices where a diagram can profitably be drawn, it is easy to distinguish reducible and irreducible elements: an element is meet-reducible if from it rise two or more connecting lines, meet-irreducible if there rises only one or none; dually, an element is join-reducible if two or more lines descend from it, join-irreducible if only one or none. It follows that the zero element and the atoms—if such elements exist—are always join-irreducible, the unity element and the dual atoms—with the same proviso—always meet-irreducible. In a finite lattice every element can be represented in at least one way and without redundancy as the join of a subset of the join-irreducible elements contained by that element.