Algebraic theory of boolean algebras

In this chapter we start to explore the theory of boolean algebras as an algebraic theory in its own right, in a way analogous to ring theory, say. We will see many applications of the Completeness and Soundness Theorems proved in the last chapter.

We start with an important definition concerning boolean algebras.

Definition 8.1 Let B, C be boolean algebras. A homomorphism from B to C is a map h: B → C such that, for all a, b ∈ B,

• h(a ν b) = h(a)νh(b)

• h(aΛb) = h(a)Λh(b)

• h(Τ) = Τ

• h(⊥) = ⊥

• h(a′) = h(a)′

Here, ν and Λ, etc., are calculated inside B on the left hand side, and inside C on the right. In fact, the last condition (on complementation) is not necessary and follows from the other four, since if those four hold then we have Τ = h(Τ) = h(a′ Λ a) = h(a′)?h(a) and ⊥ = h(⊥) = h(a′ Λa) = h(a′)Λh(a) so h(a′) = h(a)′ by Proposition 5.22 on the Uniqueness of Complements.

We will see several examples of homomorphisms later, but first we study one particular homomorphism of boolean algebras that applies to all such B.