CHAPTER 6 - DISTRIBUTIVE LATTICES

Abstract

A lattice is said to be distributive only if it satisfies several postulate. A distributive lattice cannot contain a sublattice isomorphic with the pentagonal lattice nor with the five-element partition lattice. This chapter explains that neither pentagonal nor partition lattice is distributive. The Kurosh–Ore Theorem states that if an element of a modular lattice has two irredundant representations as join of irreducibles, the same number of irreducibles appears in both. If an element of a distributive lattice can be represented as the join without redundancy of a finite set of join-irreducibles, the representation is unique. A complemented distributive lattice is called a Boolean lattice. If an element of a distributive lattice has a complement, that complement is unique. A Boolean lattice considered as algebra closed with respect to the three operations of complementation, formation of meet, formation of join, is called a Boolean algebra. A relatively complemented distributive lattice that possesses a zero element is called a generalized Boolean algebra.