Inabstract algebra, aBoolean algebra orBoolean lattice is acomplementeddistributive lattice. This type ofalgebraic structure captures essential properties of bothset operations andlogic operations. A Boolean algebra can be seen as a generalization of apower set algebra or afield of sets, or its elements can be viewed as generalizedtruth values. It is also a special case of aDe Morgan algebra and aKleene algebra (with involution).

Every Boolean algebragives rise to aBoolean ring, and vice versa, withring multiplication corresponding toconjunction ormeet ∧, and ring addition toexclusive disjunction orsymmetric difference (notdisjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while theaxioms and theorems of Boolean algebra express the symmetry of the theory described by theduality principle.^[1]

The term "Boolean algebra" honorsGeorge Boole (1815–1864), a self-educated English mathematician. He introduced thealgebraic system initially in a small pamphlet,The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy betweenAugustus De Morgan andWilliam 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 byWilliam Jevons andCharles Sanders Peirce.

The first systematic presentation of Boolean algebra anddistributive lattices is owed to the 1890Vorlesungen ofErnst Schröder. The first extensive treatment of Boolean algebra in English isA. N. Whitehead's 1898Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper byEdward V. Huntington.^[2] Boolean algebra came of age as serious mathematics with the work ofMarshall Stone in the 1930s, and withGarrett Birkhoff's 1940Lattice Theory. In the 1960s,Paul Cohen,Dana Scott, and others found deep new results inmathematical logic andaxiomatic set theory using offshoots of Boolean algebra, namelyforcing andBoolean-valued models.

ABoolean algebra is asetA, equipped with twobinary operations∧ (called "meet" or "and"),∨ (called "join" or "or"), aunary operation¬ (called "complement" or "not") and two elements0 and1 inA (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols⊥ and⊤, respectively), such that for all elementsa,b andc ofA, the followingaxioms hold:^[3]

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
a ∨ 0 =a	a ∧ 1 =a	identity
a ∨ (b ∧c) = (a ∨b) ∧ (a ∨c)	a ∧ (b ∨c) = (a ∧b) ∨ (a ∧c)	distributivity
a ∨ ¬a = 1	a ∧ ¬a = 0	complements

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 (seeProven properties).

A Boolean algebra with only one element is called atrivial Boolean algebra or adegenerate Boolean algebra. (In older works, some authors required0 and1 to bedistinct elements in order to exclude this case.)^{[citation needed]}

It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that

a =b ∧a     if and only if    a ∨b =b.

The relation≤ defined bya ≤b if these equivalent conditions hold, is apartial order with least element 0 and greatest element 1. The meeta ∧b and the joina ∨b of two elements coincide with theirinfimum andsupremum, respectively, with respect to ≤.

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

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

The set of axioms isself-dual in the sense that if one exchanges∨ with∧ and0 with1 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 itsdual.^[4]

• The simplest non-trivial Boolean algebra, thetwo-element Boolean algebra, has only two elements,0 and1, and is defined by the rules:

• It has applications inlogic, interpreting0 asfalse,1 astrue,∧ asand,∨ asor, and¬ asnot. 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 arelogically equivalent.

• The two-element Boolean algebra is also used for circuit design inelectrical engineering;^{[note 1]} here 0 and 1 represent the two different states of onebit in adigital circuit, typically high and lowvoltage. 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 trivialbrute 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:
  • (a ∨b) ∧ (¬a ∨c) ∧ (b ∨c) ≡ (a ∨b) ∧ (¬a ∨c)
  • (a ∧b) ∨ (¬a ∧c) ∨ (b ∧c) ≡ (a ∧b) ∨ (¬a ∧c)

• Thepower set (set of all subsets) of any given nonempty setS forms a Boolean algebra, analgebra of sets, with the two operations∨ := ∪ (union) and∧ := ∩ (intersection). The smallest element 0 is theempty set and the largest element1 is the setS itself.

• After the two-element Boolean algebra, the simplest Boolean algebra is that defined by thepower set of two atoms:

• The setA of all subsets ofS that are either finite orcofinite is a Boolean algebra and analgebra of sets called thefinite–cofinite algebra. IfS is infinite then the set of all cofinite subsets ofS, which is called theFréchet filter, is a freeultrafilter onA. However, the Fréchet filter is not an ultrafilter on the power set ofS.
• Starting with thepropositional calculus withκ sentence symbols, form theLindenbaum algebra (that is, the set of sentences in the propositional calculus modulological equivalence). This construction yields a Boolean algebra. It is in fact thefree Boolean algebra onκ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
• Given anylinearly ordered setL with a least element, the interval algebra is the smallest Boolean algebra of subsets ofL containing all of the half-open intervals[a,b) such thata is inL andb is either inL or equal to∞. Interval algebras are useful in the study ofLindenbaum–Tarski algebras; everycountable Boolean algebra is isomorphic to an interval algebra.

• For anynatural numbern, the set of all positivedivisors ofn, defininga ≤b ifadividesb, forms adistributive lattice. This lattice is a Boolean algebra if and only ifn issquare-free. The bottom and the top elements of this Boolean algebra are the natural numbers1 andn, respectively. The complement ofa is given byn/a. The meet and the join ofa andb are given by thegreatest common divisor (gcd) and theleast common multiple (lcm) ofa andb, respectively. The ring additiona +b is given bylcm(a,b) / gcd(a,b). The picture shows an example forn = 30. As a counter-example, considering the non-square-freen = 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 fromtopological spaces: ifX is a topological space, then the collection of all subsets ofX that areboth open and closed forms a Boolean algebra with the operations∨ := ∪ (union) and∧ := ∩ (intersection).
• IfR is an arbitrary ring then its set ofcentral idempotents, which is the set

$${\displaystyle A=\left\{e\in R:e^2=e\text{ and }ex=xe\text{ for all }x\in R\right\},}$$becomes a Boolean algebra when its operations are defined bye ∨f :=e +f −ef ande ∧f :=ef.

Ahomomorphism between two Boolean algebrasA andB is afunctionf :A →B such that for alla,b inA:

f(a ∨b) =f(a) ∨f(b),

f(a ∧b) =f(a) ∧f(b),

f(0) = 0,

f(1) = 1.

It then follows thatf(¬a) = ¬f(a) for alla inA. Theclass of all Boolean algebras, together with this notion of morphism, forms afull subcategory of thecategory of lattices.

Anisomorphism between two Boolean algebrasA andB is a homomorphismf :A →B with an inverse homomorphism, that is, a homomorphismg :B →A such that thecompositiong ∘f :A →A is theidentity function onA, and the compositionf ∘g :B →B is the identity function onB. A homomorphism of Boolean algebras is an isomorphism if and only if it isbijective.

Every Boolean algebra(A, ∧, ∨) gives rise to aring(A, +, ·) by defininga +b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨b) ∧ ¬(a ∧b) (this operation is calledsymmetric difference in the case of sets andXOR in the case of logic) anda ·b :=a ∧b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the1 of the Boolean algebra. This ring has the property thata ·a =a for alla inA; rings with this property are calledBoolean rings.

Conversely, if a Boolean ringA is given, we can turn it into a Boolean algebra by definingx ∨y :=x +y + (x ·y) andx ∧y :=x ·y.^[5][6] 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 mapf :A →B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. Thecategories of Boolean rings and Boolean algebras areequivalent;^[7] in fact the categories areisomorphic.

Hsiang (1985) gave arule-based algorithm tocheck whether two arbitrary expressions denote the same value in every Boolean ring.^[8]

More generally, Boudet,Jouannaud, and Schmidt-Schauß (1989)^[9] gave an algorithm tosolve equations between arbitrary Boolean-ring expressions. Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications inautomated theorem proving.

Anideal of the Boolean algebraA is a nonempty subsetI such that for allx,y inI we havex ∨y inI and for alla inA we havea ∧x inI. This notion of ideal coincides with the notion ofring ideal in the Boolean ringA. An idealI ofA is calledprime ifI ≠A and ifa ∧b inI always impliesa inI orb inI. Furthermore, for everya ∈A we have thata ∧ −a = 0 ∈I, and then ifI is prime we havea ∈I or−a ∈I for everya ∈A. An idealI ofA is calledmaximal ifI ≠A and if the only ideal properly containingI isA itself. For an idealI, ifa ∉I and−a ∉I, thenI ∪ {a} orI ∪ {−a} is contained in another proper idealJ. Hence, such anI 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 ofprime ideal andmaximal ideal in the Boolean ringA.

The dual of anideal is afilter. Afilter of the Boolean algebraA is a nonempty subsetp such that for allx,y inp we havex ∧y inp and for alla inA we havea ∨x inp. The dual of amaximal (orprime)ideal in a Boolean algebra isultrafilter. Ultrafilters can alternatively be described as2-valued morphisms fromA to the two-element Boolean algebra. The statementevery filter in a Boolean algebra can be extended to an ultrafilter is called theultrafilter lemma and cannot be proven inZermelo–Fraenkel set theory (ZF), ifZF isconsistent. Within ZF, the ultrafilter lemma is strictly weaker than theaxiom of choice. The ultrafilter lemma has many equivalent formulations:every Boolean algebra has an ultrafilter,every ideal in a Boolean algebra can be extended to a prime ideal, etc.

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It can be shown that everyfinite 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 apower of two.

Stone's celebratedrepresentation theorem for Boolean algebras states thatevery Boolean algebraA is isomorphic to the Boolean algebra of allclopen sets in some (compacttotally disconnectedHausdorff) topological space.

The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematicianAlfred North Whitehead in 1898.^[10][11] It included theabove axioms and additionallyx ∨ 1 = 1 andx ∧ 0 = 0. In 1904, the American mathematicianEdward V. Huntington (1874–1952) gave probably the most parsimonious axiomatization based on∧,∨,¬, even proving the associativity laws (see box).^[12] He also proved that these axioms areindependent of each other.^[13]

In 1933, Huntington set out the following elegant axiomatization for Boolean algebra.^[14] It requires just one binary operation+ and aunary functional symboln, 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) aRobbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as theRobbins conjecture) remained open for decades, and became a favorite question ofAlfred Tarski and his students.

In 1996,William McCune atArgonne 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 computer programEQP he designed. For a simplification of McCune's proof, see Dahn (1998).^[15]

Further work has been done for reducing the number of axioms; seeMinimal axioms for Boolean algebra.

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Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, adistributive latticeB is a generalized Boolean lattice, if it has a smallest element0 and for any elementsa andb inB such thata ≤b, there exists an elementx such thata ∧x = 0 anda ∨x =b. Defininga \b as the uniquex such that(a ∧b) ∨x =a and(a ∧b) ∧x = 0, we say that the structure(B, ∧, ∨, \, 0) is ageneralized Boolean algebra, while(B, ∨, 0) is ageneralized Booleansemilattice. Generalized Boolean lattices are exactly theideals of Boolean lattices.

A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called anorthocomplemented lattice. Orthocomplemented lattices arise naturally inquantum logic as lattices ofclosedlinear subspaces forseparableHilbert spaces.

• Davey, B.A.;Priestley, H.A. (1990).Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
• Cohn, Paul M. (2003),Basic Algebra: Groups, Rings, and Fields, Springer, pp. 51,70–81,ISBN 9781852335878
• Givant, Steven;Halmos, Paul (2009),Introduction to Boolean Algebras,Undergraduate Texts in Mathematics,Springer,ISBN 978-0-387-40293-2.
• Goodstein, R. L. (2012),"Chapter 2: The self-dual system of axioms",Boolean Algebra, Courier Dover Publications, pp. 21ff,ISBN 9780486154978
• Huntington, Edward V. (1904)."Sets of Independent Postulates for the Algebra of Logic".Transactions of the American Mathematical Society.5 (3):288–309.doi:10.1090/s0002-9947-1904-1500675-4.JSTOR 1986459.
• Padmanabhan, Ranganathan; Rudeanu, Sergiu (2008),Axioms for lattices and boolean algebras, World Scientific,ISBN 978-981-283-454-6.
• Stone, Marshall H. (1936)."The Theory of Representations for Boolean Algebra".Transactions of the American Mathematical Society.40:37–111.doi:10.1090/s0002-9947-1936-1501865-8.
• Whitehead, A.N. (1969) [1898].A Treatise on Universal Algebra. Cambridge University Press.ISBN 978-1-4297-0032-0.

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• Brown, Stephen; Vranesic, Zvonko (2002),Fundamentals of Digital Logic with VHDL Design (2nd ed.),McGraw–Hill,ISBN 978-0-07-249938-4. See Section 2.5
• Cori, Rene; Lascar, Daniel (2000),Mathematical Logic: A Course with Exercises,Oxford University Press,ISBN 978-0-19-850048-3. See Chapter 2.

• Halmos, Paul (1963),Lectures on Boolean Algebras, Van Nostrand.
• Halmos, Paul; Givant, Steven (1998),Logic as Algebra, Dolciani Mathematical Expositions, vol. 21,Mathematical Association of America,ISBN 978-0-88385-327-6.
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• Huntington, E. V. (1933b), "Boolean algebra: A correction",Transactions of the American Mathematical Society,35 (2):557–558,doi:10.2307/1989783,JSTOR 1989783
• Mendelson, Elliott (1970),Boolean Algebra and Switching Circuits, Schaum's Outline Series in Mathematics,McGraw–Hill,ISBN 978-0-07-041460-0
• Monk, J. Donald; Bonnet, R., eds. (1989),Handbook of Boolean Algebras,North-Holland,ISBN 978-0-444-87291-3. In 3 volumes. (Vol.1:ISBN 978-0-444-70261-6, Vol.2:ISBN 978-0-444-87152-7, Vol.3:ISBN 978-0-444-87153-4)
• Sikorski, Roman (1966),Boolean Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete,Springer Verlag.
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• "Boolean algebra",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Stanford Encyclopedia of Philosophy: "The Mathematics of Boolean Algebra", by J. Donald Monk.
• McCune W., 1997.Robbins Algebras Are Boolean JAR 19(3), 263–276
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• Weisstein, Eric W."Boolean Algebra".MathWorld.

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