Abstract Algebra Theory and Applications - Computer Science ...

302 CHAPTER 17 LATTICES AND BOOLEAN ALGEBRASProof. We will prove only (2). The rest of the identities are left as exercises.For a ∨ b = a ∨ c and a ∧ b = a ∧ c, we haveb = b ∨ (b ∧ a)= b ∨ (a ∧ b)= b ∨ (a ∧ c)= (b ∨ a) ∧ (b ∨ c)= (a ∨ b) ∧ (b ∨ c)= (a ∨ c) ∧ (b ∨ c)= (c ∨ a) ∧ (c ∨ b)= c ∨ (a ∧ b)= c ∨ (a ∧ c)= c ∨ (c ∧ a)= c.□Finite Boolean AlgebrasA Boolean algebra is a finite Boolean algebra if it contains a finite numberof elements as a set. Finite Boolean algebras are particularly nice since wecan classify them up to isomorphism.Let B and C be Boolean algebras. A bijective map φ : B → C is anisomorphism of Boolean algebras ifφ(a ∨ b) = φ(a) ∨ φ(b)φ(a ∧ b) = φ(a) ∧ φ(b)for all a and b in B.We will show that any finite Boolean algebra is isomorphic to the Booleanalgebra obtained by taking the power set of some finite set X. We will needa few lemmas and definitions before we prove this result. Let B be a finiteBoolean algebra. An element a ∈ B is an atom of B if a ≠ O and a ∧ b = afor all b ∈ B. Equivalently, a is an atom of B if there is no nonzero b ∈ Bdistinct from a such that O ≼ b ≼ a.Lemma 17.7 Let B be a finite Boolean algebra. If b is a nonzero elementof B, then there is an atom a in B such that a ≼ b.