Boolean Algebra Definition

The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.

The nodes N_i of a Boolean lattice can be labeled with Boolean formulae F(N_i), such that if node C is the meet of nodes A and B, then F(C) = F(A)F(B); if node C is the join of nodes A and B, then F(C) = F(A)+F(B); if node C is the complement of node A, then F(C) = F(A)'; and the '≤' order relation corresponds to logical entailment. A set of 'n' Boolean formulae could be called a "basis" for a 2ⁿ-element Boolean algebra iff they are all mutually disjoint (i.e., the product of any pair is 0) and their Σ (collective sum) is equal to 1. A set S of formulae could then generate a Boolean algebra inductively as follows: (base step) let P₀ = S ∪ {(ΣS)'}, (inductive step) if a pair of formulae F and G in P_i are non-disjoint (i.e., FG≠0), then let P_i+1 = (P_i ∪ {FG, F'G, FG'}) \ {F, G}, otherwise P_i is a basis. If CARD(P_i)=n then the Boolean algebra will have 2ⁿ elements which are all "linear combinations" of the basis elements, with a coefficient of either 0 or 1 for each term of each linear combination.