# Boolean algebra

## English

### Noun

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1. (algebra) A De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.
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 Ni of a Boolean lattice can be labeled with Boolean formulae F(Ni), 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 Pi are non-disjoint (i.e., FG≠0), then let Pi+1 = (Pi ∪ {FG, F'G, FG'}) \ {F, G}, otherwise Pi is a basis. If CARD(Pi)=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.
2. (algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT
3. (mathematics) The study of such algebras; Boolean logic, classical logic.