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In , a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and operations. A Boolean algebra can be seen as a generalization of a 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 , and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to 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 , 2009, p. 20


History
The term "Boolean algebra" honors (1815–1864), a self-educated English mathematician. He introduced the initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of in the 1930s, and with 's 1940 Lattice Theory. In the 1960s, Paul Cohen, , and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.


Definition
A Boolean algebra is a six- consisting of a set A, equipped with two ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a ¬ (called "complement" or "not") and two elements 0 and 1 (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ⊤, respectively), such that for all elements a, b and c of A, the following hold:Davey, Priestley, 1990, p.109, 131, 144

:>
a ∨ ( bc) = ( ab) ∨ c a ∧ ( bc) = ( ab) ∧ c
ab = baab = ba
a ∨ ( ab) = aa ∧ ( ab) = a
a ∨ 0 = aa ∧ 1 = a
a ∨ ( bc) = ( ab) ∧ ( ac)  a ∧ ( bc) = ( ab) ∨ ( ac)  
a ∨ ¬ a = 1a ∧ ¬ a = 0complements
Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties).

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

: a = ba     if and only if     ab = b.
The relation ≤ defined by ab if these equivalent conditions hold, is a with least element 0 and greatest element 1. The meet ab and the join ab of two elements coincide with their and , respectively, with respect to ≤.

The first four pairs of axioms constitute a definition of a .

It follows from the first five pairs of axioms that any complement is unique.

The set of axioms is self-dual 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..


Examples
  • The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, 0 and 1, and is defined by the rules:
{ class="wikitable" border="1" cellpadding="4" cellspacing="0"
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* It has applications in , interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.

* The two-element Boolean algebra is also used for circuit design in electrical engineeringStrictly, electrical engineers tend to use additional states to represent other circuit conditions such as high impedance - see IEEE 1164 or IEEE 1364.; here 0 and 1 represent the two different states of one in a , typically high and low . Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.

* The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws ( Consensus theorems) are generally valid in all Boolean algebras:
** ( ab) ∧ (¬ ac) ∧ ( bc) ≡ ( ab) ∧ (¬ ac)
** ( ab) ∨ (¬ ac) ∨ ( bc) ≡ ( ab) ∨ (¬ ac)

  • The (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the and the largest element 1 is the set S itself.

* After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the of two atoms:
{ class="wikitable" border="1" cellpadding="4" cellspacing="0"
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| | width="40" | |
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  • The set of all subsets of S that are either finite or is a Boolean algebra, an algebra of sets.
  • Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo logical equivalence). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
  • Given any set L with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the half-open intervals [ a, b) such that a is in L and b is either in L or equal to ∞. Interval algebras are useful in the study of Lindenbaum–Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.

  • For any n, the set of all positive of n, defining ab if a b, forms a distributive lattice. This lattice is a Boolean algebra if and only if n is square-free. The bottom and the top element of this Boolean algebra is the natural number 1 and n, respectively. The complement of a is given by n/ a. The meet and the join of a and b is given by the greatest common divisor (gcd) and the least common multiple (lcm) of a and b, respectively. The ring addition a+ b is given by lcm( a, b)/gcd( a, b). The picture shows an example for n = 30. As a counter-example, considering the non-square-free n=60, the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
  • Other examples of Boolean algebras arise from : if X is a topological space, then the collection of all subsets of X which are forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
  • If R is an arbitrary ring and we define the set of central idempotents by
    A = { eR : e2 = e, ex = xe, ∀ xR }
    then the set A becomes a Boolean algebra with the operations ef := e + f - ef and ef := ef.


Homomorphisms and isomorphisms
A between two Boolean algebras A and B is a function f : AB such that for all a, b in A:

f( ab) = f( a) ∨ f( b),
f( ab) = f( a) ∧ f( b),
f(0) = 0,
f(1) = 1.

It then follows that fa) = ¬ f( a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a of the of lattices.

An isomorphism between two Boolean algebras A and B is a homomorphism f : AB with an inverse homomorphism, that is, a homomorphism g : BA such that the composition gf: AA is the identity function on A, and the composition fg: BB is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is .


Boolean rings
Every Boolean algebra (A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b) ∨ ( b ∧ ¬ a) = ( ab) ∧ ¬( ab) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a · b := ab. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called .

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining xy := x + y + ( x · y) and xy := 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 : AB is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The of Boolean rings and Boolean algebras are equivalent., p. 81.

Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring.

More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.


Ideals and filters
An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have ax in I. This notion of ideal coincides with the notion of in the Boolean ring A. An ideal I of A is called prime if IA and if ab in I always implies a in I or b in I. Furthermore, for every aA we have that a-a = 0 ∈ I and then aI or -aI for every aA, if I is prime. An ideal I of A is called maximal if IA and if the only ideal properly containing I is A itself. For an ideal I, if aI and -aI, then I ∪ { a} or I ∪ { -a} is properly contained in another ideal J. Hence, that an I is not maximal and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of and in the Boolean ring A.

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 xy in p and for all a in A we have ax in p. The dual of a maximal (or prime) ideal in a Boolean algebra is . Ultrafilters can alternatively be described as 2-valued morphisms from A to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and can not be proved in ZF, if ZF is . 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.


Representations
It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.

Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all sets in some ( totally disconnected ) topological space.


Axiomatics
{ align="left" class="collapsible collapsed" style="text-align:left" ! UId1 !! !! colspan="2" If xo = x for all x, then o = 0
If xo = x, then
0
by assumption
by Cmm1
by Idn1
| UId2   dual   If xi = x for all x, then i = 1 |- valign="top" |
xx
by Idn2
by Cpl1
by Dst1
by Cpl2
by Idn1
| Idm2   dual   xx = x |- valign="top" |
x ∨ 1
by Idn2
by Cmm2
by Cpl1
by Dst1
by Idn2
by Cpl1
| Bnd2   dual   x ∧ 0 = 0 |- valign="top" |
x ∨ ( xy)
by Idn2
by Dst2
by Cmm1
by Bnd1
by Idn2
| Abs2   dual   x ∧ ( xy) = x |- valign="top" | colspan="2" |
If xxn = 1 and xxn = 0, then
xn
by Idn2
by Cpl1
by Dst2
by Cmm2
by assumption
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by Dst2
by assumption
by Idn2
|- valign="top" | colspan="2" |
by Cmm1, Cpl1
by Cmm2, Cpl2
by UNg
|- valign="top" |
x ∨ (¬ xy)
by Idn2
by Cmm2
by Cpl1
by Dst1
by Abs2
by Cpl1
| A2   dual   x ∧ (¬ xy) = 0 |- valign="top" |
( xy) ∨ (¬ x ∧ ¬ y)
by Dst1
by Cmm1
by DNg
by A1
by Idn2
| B2   dual   ( xy) ∧ (¬ x ∨ ¬ y) = 0 |- valign="top" |
( xy) ∧ (¬ x ∧ ¬ y)
by Cmm2
by Dst2
by Cmm2
by A2
by Idn1
| C2   dual   ( xy) ∨ (¬ x ∨ ¬ y) = 1 |- valign="top" |
by B1, C1, and UNg
| DMg2   dual   ¬( xy) = ¬ x ∨ ¬ y |- valign="top" |
( x ∨ ( yz)) ∨ ¬ x
by Cmm1
by DNg
by A1
| D2   dual   ( x∧( yz)) ∧ ¬ x = 0 |- valign="top" |
y ∧ ( x ∨ ( yz))
by Dst2
by Abs2
by Cmm1
by Abs1
| E2   dual   y ∨ ( x∧( yz)) = y |- valign="top" |
( x ∨ ( yz)) ∨ ¬ y
by Cmm1
by Idn2
by Cmm2
by Cpl1
by Cmm1
by Dst1
by E1
by Cmm1
by Cpl1
| F2   dual   ( x∧( yz)) ∧ ¬ y = 0 |- valign="top" |
( x ∨ ( yz)) ∨ ¬ z
by Cmm1
by F1
| G2   dual   ( x∧( yz)) ∧ ¬ z = 0 |- valign="top" |
¬(( xy) ∨ z) ∧ x
by DMg1
by DMg1
by Cmm2
by Idn1
by Cmm1
by Cpl1
by Dst2
by Cmm2
by E2
by Cpl2
| H2   dual   ¬(( xy)∧ z) ∨ x = 1 |- valign="top" |
¬(( xy) ∨ z) ∧ y
by Cmm1
by H1
| I2   dual   ¬(( xy)∧ z) ∨ y = 1 |- valign="top" |
¬(( xy) ∨ z) ∧ z
by DMg1
by Cmm2
by Cmm2
by A2
| J2   dual   ¬(( xy)∧ z) ∨ z = 1 |- valign="top" |
( x∨( yz)) ∨ ¬(( xy) ∨ z)
by DMg1
by DMg1
by Dst1
by Dst1
by D1, F1, G1
by Idn2
| K2   dual   ( x ∧ ( yz)) ∧ ¬(( xy) ∧ z) = 0 |- valign="top" |
( x ∨ ( yz)) ∧ ¬(( xy) ∨ z)
by Cmm2
by Dst2
by Dst2
by H1, I1, J1
by Idn1
| L2   dual   ( x ∧ ( yz)) ∨ ¬(( xy) ∧ z) = 1 |- valign="top" |
by K1, L1, UNg, DNg
| Ass2   dual   x ∧ ( yz) = ( xy) ∧ z |- | colspan="2" |
Unique Identity
Unique Negation
De Morgan's Law
|}
x ∨ 0 = xx ∧ 1 = x
xy = yxxy = yx
x ∨ ( yz) = ( xy) ∧ ( xz)x ∧ ( yz) = ( xy) ∨ ( xz)
x ∨ ¬ x = 1x ∧ ¬ x = 0
{ align="left" class="collapsible" style="text-align:left"
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.292-293, (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:

  1. Commutativity: x + y = y + x.
  2. Associativity: ( x + y) + z = x + ( y + z).
  3. Huntington equation: n( n( x) + y) + n( n( x) + n( y)) = x.

immediately asked: If the Huntington equation is replaced with its dual, to wit:

4. Robbins Equation: n( n( x + y) + n( x + n( y))) = x,

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 and his students. In 1996, 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 he designed. For a simplification of McCune's proof, see Dahn (1998).


Generalizations
Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that ab, there exists an element x such that a ∧ x = 0 and a ∨ x = b. Defining a ∖ b as the unique x such that (a ∧ b) ∨ x = a and (a ∧ b) ∧ x = 0, we say that the structure (B,∧,∨,∖,0) is a generalized Boolean algebra, while (B,∨,0) is a generalized Boolean . Generalized Boolean lattices are exactly the ideals of Boolean lattices.

A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in as lattices of closed subspaces for separable .


See also


Notes
  • . See Section 2.5.
  • . See Chapter 2.
  • .
  • .
  • .
  • .
  • .
  • .
  • .
  • . In 3 volumes. (Vol.1:, Vol.2:, Vol.3:)
  • .
  • .
  • . Reprinted by Dover Publications, 1979.
  • (1898). 9781429700320, Cambridge University Press. .


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