11 Boolean Algebras and Lattices
The definition of a Boolean algebra did not mention the inclusion relation, even though this is the most fundamental concept of all. This chapter replaces haphazard manipulation of Boolean polynomials by a systematic procedure. The truth or falsity of any purported equation E1 = E2 in Boolean algebra can be settled definitely, simply by reducing each side to disjunctive canonical form. The consistency principle shows how to define inclusion in terms of join or meet. The chapter shows that, conversely, one can define join and meet in terms of inclusion. The chapter proves a stronger result, showing in passing that the postulates used to define distributive lattices completely characterize the properties of intersections and unions of sets. The main conclusion is that the postulates which were assumed for Boolean algebra imply all true identities for the algebra of sets with respect to intersection, union, and complement.