In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized . It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive or or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.Givant and Paul Halmos, 2009, p. 20
a ∨ ( b ∨ c) = ( a ∨ b) ∨ c  a ∧ ( b ∧ c) = ( a ∧ b) ∧ c  associativity 
a ∨ b = b ∨ a  a ∧ b = b ∧ a  commutativity 
a ∨ ( a ∧ b) = a  a ∧ ( a ∨ b) = a  Absorption law 
a ∨ 0 = a  a ∧ 1 = a  identity element 
a ∨ ( b ∧ c) = ( a ∨ b) ∧ ( a ∨ c)  a ∧ ( b ∨ c) = ( a ∧ b) ∨ ( a ∧ c)  distributivity 
a ∨ ¬ a = 1  a ∧ ¬ a = 0  complements 
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)
It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
The first four pairs of axioms constitute a definition of a bounded lattice.
It follows from the first five pairs of axioms that any complement is unique.
The set of axioms is selfdual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual..
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It then follows that f(¬ a) = ¬ f( a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category theory of lattices.
An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the composition g ◌ f: A → A is the identity function on A, and the composition f ◌ g: B → B is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijection.
Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y) and x ∧ y := x · y. Stone, 1936Hsiang, 1985, p.260 Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The category theory of Boolean rings and Boolean algebras are equivalent., p. 81.
Hsiang (1985) gave a rulebased algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring.
More generally, Boudet, Jouannaud, and SchmidtSchauß (1989) gave an algorithm to solve equations between arbitrary Booleanring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.
The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x ∧ y in p and for all a in A we have a ∨ x in p. The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2valued morphisms from A to the twoelement Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and cannot be proven in ZF, if ZF is consistent. Within ZF, it is strictly weaker than the axiom of choice. The Ultrafilter Theorem has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal, etc.
Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all clopen set sets in some (compact space totally disconnected Hausdorff space) topological space.
{ align="left" class="collapsible collapsed" style="textalign:left" ! UId_{1} !! !! colspan="2"  If x ∨ o = x for all x, then o = 0 
If x ∨ o = x, then  
0  
by assumption  
by Cmm_{1}  
by Idn_{1} 
x ∨ x 
by Idn_{2} 
by Cpl_{1} 
by Dst_{1} 
by Cpl_{2} 
by Idn_{1} 
x ∨ 1 
by Idn_{2} 
by Cmm_{2} 
by Cpl_{1} 
by Dst_{1} 
by Idn_{2} 
by Cpl_{1} 
x ∨ ( x ∧ y) 
by Idn_{2} 
by Dst_{2} 
by Cmm_{1} 
by Bnd_{1} 
by Idn_{2} 
If x ∨ x_{n} = 1 and x ∧ x_{n} = 0, then  
x_{n}  
by Idn_{2}  
by Cpl_{1}  
by Dst_{2}  
by Cmm_{2}  
by assumption  
by Cpl_{2}  
by Cmm_{2}  
by Dst_{2}  
by assumption  
by Idn_{2} 
by Cmm_{1}, Cpl_{1} 
by Cmm_{2}, Cpl_{2} 
by UNg 
x ∨ (¬ x ∨ y) 
by Idn_{2} 
by Cmm_{2} 
by Cpl_{1} 
by Dst_{1} 
by Abs_{2} 
by Cpl_{1} 
( x ∨ y) ∨ (¬ x ∧ ¬ y) 
by Dst_{1} 
by Cmm_{1} 
by DNg 
by A_{1} 
by Idn_{2} 
( x ∨ y) ∧ (¬ x ∧ ¬ y) 
by Cmm_{2} 
by Dst_{2} 
by Cmm_{2} 
by A_{2} 
by Idn_{1} 
by B_{1}, C_{1}, and UNg 
( x ∨ ( y ∨ z)) ∨ ¬ x 
by Cmm_{1} 
by DNg 
by A_{1} 
y ∧ ( x ∨ ( y ∨ z)) 
by Dst_{2} 
by Abs_{2} 
by Cmm_{1} 
by Abs_{1} 
( x ∨ ( y ∨ z)) ∨ ¬ y 
by Cmm_{1} 
by Idn_{2} 
by Cmm_{2} 
by Cpl_{1} 
by Cmm_{1} 
by Dst_{1} 
by E_{1} 
by Cmm_{1} 
by Cpl_{1} 
( x ∨ ( y ∨ z)) ∨ ¬ z 
by Cmm_{1} 
by F_{1} 
¬(( x ∨ y) ∨ z) ∧ x 
by DMg_{1} 
by DMg_{1} 
by Cmm_{2} 
by Idn_{1} 
by Cmm_{1} 
by Cpl_{1} 
by Dst_{2} 
by Cmm_{2} 
by E_{2} 
by Cpl_{2} 
¬(( x ∨ y) ∨ z) ∧ y 
by Cmm_{1} 
by H_{1} 
¬(( x ∨ y) ∨ z) ∧ z 
by DMg_{1} 
by Cmm_{2} 
by Cmm_{2} 
by A_{2} 
( x∨( y∨ z)) ∨ ¬(( x ∨ y) ∨ z) 
by DMg_{1} 
by DMg_{1} 
by Dst_{1} 
by Dst_{1} 
by D_{1}, F_{1}, G_{1} 
by Idn_{2} 
( x ∨ ( y ∨ z)) ∧ ¬(( x ∨ y) ∨ z) 
by Cmm_{2} 
by Dst_{2} 
by Dst_{2} 
by H_{1}, I_{1}, J_{1} 
by Idn_{1} 
by K_{1}, L_{1}, UNg, DNg 
Unique Identity 
Idempotence 
Bounded lattice 
Absorption law 
Unique Negation 
Double negation 
De Morgan's Law 
Associativity 
x ∨ 0 = x  x ∧ 1 = x 
x ∨ y = y ∨ x  x ∧ y = y ∧ x 
x ∨ ( y∧ z) = ( x∨ y) ∧ ( x∨ z)  x ∧ ( y∨ z) = ( x∧ y) ∨ ( x∧ z) 
x ∨ ¬ x = 1  x ∧ ¬ x = 0 
{ align="left" class="collapsible" style="textalign:left"  
Identity element  
Commutativity  
Distributivity  
Complements 
The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898.Padmanabhan, p. 73Whitehead, 1898, p.37 It included the above axioms and additionally x∨1=1 and x∧0=0. In 1904, the American mathematician Edward V. Huntington (1874–1952) gave probably the most parsimonious axiomatization based on ∧, ∨, ¬, even proving the associativity laws (see box).Huntington, 1904, p.292293, (first of several axiomatizations by Huntington) He also proved that these axioms are independent of each other.Huntington, 1904, p.296 In 1933, Huntington set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation + and a unary functional symbol n, to be read as 'complement', which satisfy the following laws:
Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:
do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students. In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the automated reasoning program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).
A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable .

