Chapter 13 - Boolean Algebras

Abstract

This chapter provides an overview on the boolean algebras. A boolean lattice is a complemented lattice that is also distributive. This lattice is complete if its ordering is complete—that is, if every subset S ⊆ X has a supremum and an infimum. A Boolean lattice is essentially the same thing as a Boolean algebra, and the two terms may be used interchangeably. Although Boolean rings and Boolean lattices are not the same, there is a natural correspondence between them, and for that reason the terms “Boolean rings” and “Boolean lattices” are sometimes used interchangeably. Boolean lattices are not much more general than algebras of sets. In fact, the Stone Representation Theorem states that every Boolean algebra is isomorphic to some algebra of sets. Aside from the conceptual difficulty of intangibles, the chief difference between Boolean lattices and algebras of sets is one of viewpoint. When considering algebras of sets, the chapter considers the points that make up these sets. In contrast, the members of a Boolean lattice are considered as elements, not necessarily containing any “points.”