Boolean algebra (structure): Difference between revisions

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{{short description|Algebraic structure modeling logical operations}}

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{{for-multi|an introduction to the subject|Boolean algebra|an alternative presentation|Boolean algebras canonically defined}}

<div style="float: left;">[[Image:Wikibooks-logo-en.png|50px|none|Wikibooks]]</div>

{{Use dmy dates|date=November 2020}}

<div style="margin-left: 60px;">[[wikibooks:|Wikibooks ]] has more about '''''[[Boolean logic]]''''', under the somewhat misleading title

In [[abstract algebra]], a '''Boolean algebra''' or '''Boolean lattice''' is a [[complemented lattice|complemented]] [[distributive lattice]]. This type of [[algebraic structure]] captures essential properties of both [[Set (mathematics)|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 [[truth value]]s. It is also a special case of a [[De Morgan algebra]] and a [[Kleene algebra (with involution)]].

<div style="margin-left: 10px;">'''''[[wikibooks::Boolean Algebra|Boolean Algebra]]'''''</div>

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:''For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see [[Boolean logic]].''

:''For the use of binary numbers in computer systems, please see the article [[binary arithmetic]].''

Every Boolean algebra [[#Boolean rings|gives rise]] to a [[Boolean ring]], and vice versa, with [[ring (mathematics)|ring]] multiplication corresponding to [[logical conjunction|conjunction]] or [[meet (mathematics)|meet]] ∧, and ring addition to [[exclusive or|exclusive disjunction]] or [[symmetric difference]] (not [[logical disjunction|disjunction]] ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the [[axiom]]s and theorems of Boolean algebra express the symmetry of the theory described by the [[Duality principle (Boolean algebra)|duality principle]].{{sfn|Givant|Halmos|2009|p=20}}

In [[abstract algebra]], a '''Boolean algebra''' is an [[algebraic structure]] (a collection of elements and operations on them obeying defining [[axiom#Non-logical axioms|axioms]]) that captures essential properties of both [[set]] operations and [[logic]] operations. Specifically, it deals with the [[set]] operations of [[intersection (set theory)|intersection]], [[union (set theory)|union]], [[complement (set theory)|complement]]; and the [[logic]] operations of [[logical conjunction|AND]], [[logical disjunction|OR]], [[logical negation|NOT]].

[[Image:Hasse diagram of powerset of 3.svg|thumb|right|250px|Boolean lattice of subsets]]

__TOC__

== History ==

For example, the logical assertion that a statement ''a'' and its negation ¬''a'' cannot both be true,

<!-- [[Boolean algebra (history)]] redirects here -->

The term "Boolean algebra" honors [[George Boole]] (1815–1864), a self-educated English mathematician. He introduced the [[algebraic system]] 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 [[Sir William Hamilton, 9th Baronet|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 [[William Jevons]] and [[Charles Sanders Peirce]]. The first systematic presentation of Boolean algebra and [[distributive lattice]]s is owed to the 1890 ''Vorlesungen'' of [[Ernst Schröder (mathematician)|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 [[Marshall Stone]] in the 1930s, and with [[Garrett Birkhoff]]'s 1940 ''Lattice Theory''. In the 1960s, [[Paul Cohen (mathematician)|Paul Cohen]], [[Dana Scott]], and others found deep new results in [[mathematical logic]] and [[axiomatic set theory]] using offshoots of Boolean algebra, namely [[forcing (mathematics)|forcing]] and [[Boolean-valued model]]s.

[[Image:Hasse diagram of powerset of 3.png|right|thumb|250px|Boolean lattice of subsets]]

:<math>a\land(\lnot a) = \mbox{FALSE},</math>

== Definition ==

parallels the set-theory assertion that a subset ''A'' and its complement ''A''^''C'' have empty intersection,

A '''Boolean algebra''' is a [[Set (mathematics)|set]] {{math|1=''A''}}, equipped with two [[binary operation]]s {{math|1=∧}} (called "meet" or "and"), {{math|1=∨}} (called "join" or "or"), a [[unary operation]] {{math|1=¬}} (called "complement" or "not") and two elements {{math|1=0}} and {{math|1=1}} in {{math|1=''A''}} (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols {{math|1=⊥}} and {{math|1=⊤}}, respectively), such that for all elements {{math|''a''}}, {{math|''b''}} and {{math|''c''}} of {{math|''A''}}, the following [[axiom]]s hold:{{sfn|Davey|Priestley|1990|pp=109, 131, 144}}

:<math>A\cap(A^C) = \empty.</math>

:: {| cellpadding=5

Because truth values can be represented as [[binary numeral system|binary numbers]] or as voltage levels in [[logic circuit]]s, the parallel extends to these as well. Thus the theory of Boolean algebras has many practical applications in [[electrical engineering]] and [[computer science]], as well as in [[mathematical logic]].

|{{math|1=''a'' ∨ (''b'' ∨ ''c'') = (''a'' ∨ ''b'') ∨ ''c''}}

|{{math|1=''a'' ∧ (''b'' ∧ ''c'') = (''a'' ∧ ''b'') ∧ ''c''}}

| [[associativity]]

|-

|{{math|1=''a'' ∨ ''b'' = ''b'' ∨ ''a''}}

|{{math|1=''a'' ∧ ''b'' = ''b'' ∧ ''a''}}

| [[commutativity]]

|-

|{{math|1=''a'' ∨ (''a'' ∧ ''b'') = ''a''}}

|{{math|1=''a'' ∧ (''a'' ∨ ''b'') = ''a''}}

| [[Absorption law|absorption]]

|-

|{{math|1=''a'' ∨ 0 = ''a''}}

|{{math|1=''a'' ∧ 1 = ''a''}}

| [[identity element|identity]]

|-

|{{math|1=''a'' ∨ (''b'' ∧ ''c'') = (''a'' ∨ ''b'') ∧ (''a'' ∨ ''c'')&nbsp;&nbsp;}}

|{{math|1=''a'' ∧ (''b'' ∨ ''c'') = (''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'')&nbsp;&nbsp;}}

| [[distributivity]]

|-

|{{math|1=''a'' ∨ ¬''a'' = 1}}

|{{math|1=''a'' ∧ ¬''a'' = 0}}

| [[complemented lattice|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 (see [[#Axiomatics|Proven properties]]).

A Boolean algebra with only one element is called a '''trivial Boolean algebra''' or a '''degenerate Boolean algebra'''. (In older works, some authors required {{math|0}} and {{math|1}} to be ''distinct'' elements in order to exclude this case.){{citation needed|date=July 2020}}

A Boolean algebra is also called a '''Boolean lattice'''. The connection to [[lattice (order)|lattice]]s (special [[partially ordered set]]s) is suggested by the parallel between set [[subset|inclusion]], ''A''&nbsp;⊆&nbsp;''B'', and [[order theory|ordering]], ''a''&nbsp;&le;&nbsp;''b''. Consider the lattice of all subsets of {''x'',''y'',''z''}, ordered by set inclusion. This Boolean lattice is a partially ordered set in which, say, {''x''} &nbsp;&le;&nbsp;{''x'',''y''}. Any two lattice elements, say ''p''&nbsp;= {''x'',''y''} and ''q''&nbsp;= {''y'',''z''}, have a least upper bound, here {''x'',''y'',''z''}, and a greatest lower bound, here {''y''}. Suggestively, the least upper bound (or join or supremum) is denoted by the same symbol as logical OR, ''p''∨''q''; and the greatest lower bound (or meet or infimum) is denoted by same symbol as logical AND, ''p''∧''q''.

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

The lattice interpretation helps in generalizing to [[Heyting algebra]]s, which are Boolean algebras freed from the restriction that either a statement or its negation must be true. Heyting algebras correspond to [[intuitionistic logic|intuitionist (constructivist) logic]] just as Boolean algebras correspond to [[classical logic]].

: {{math|1=''a'' = ''b'' ∧ ''a''}} &nbsp;&nbsp;&nbsp; if and only if &nbsp;&nbsp;&nbsp; {{math|1=''a'' ∨ ''b'' = ''b''}}.

The relation {{math|≤}} defined by {{math|''a'' ≤ ''b''}} if these equivalent conditions hold, is a [[partial order]] with least element 0 and greatest element 1. The meet {{math|1=''a'' ∧ ''b''}} and the join {{math|1=''a'' ∨ ''b''}} of two elements coincide with their [[infimum]] and [[supremum]], respectively, with respect to ≤.

The first four pairs of axioms constitute a definition of a [[bounded lattice]].

Boolean algebras are named after [[George Boole]], an English mathematician at [[University College Cork]]. The algebraic system of logic Boole formulated is distinct from that described in this article in some small but important respects.

__TOC__

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

die

The set of axioms is [[duality (order theory)|self-dual]] in the sense that if one exchanges {{math|1=∨}} with {{math|1=∧}} and {{math|0}} with {{math|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'''.{{sfn|Goodstein|2012|p=21ff}}

== Examples ==

* The simplest non-trivial Boolean algebra, the [[two-element Boolean algebra]], has only two elements, {{math|0}} and {{math|1}}, and is defined by the rules:

{|

|-

| width="8070" |

|

{| class="wikitable"

{| border="1" cellpadding="4" cellspacing="0"

|-

! &and;{{math|1=∧}} || {{math|0}} || {{math|1}}

|-

! {{math|0}}

| {{math|0}} || {{math|0}}

|-

! {{math|1}}

| {{math|0}} || {{math|1}}

|}

| width="40" |

|

| width="30" |

{| border="1" cellpadding="4" cellspacing="0"

|

{| class="wikitable"

|-

! &or;{{math|1=∨}} || {{math|0}} || {{math|1}}

|-

! {{math|0}}

| {{math|0}} || {{math|1}}

|-

! {{math|1}}

| {{math|1}} || {{math|1}}

|}

|

| width="40" |

|

{| class="wikitable"

|-

! {{math|''a''}} || {{math|0}} || {{math|1}}

! 0

| 0 || 1

|-

! {{math|¬''a''}}

! 1

| {{math|1}} || 1{{math|0}}

|}

|}

:* It has applications in [[logic]], interpreting {{math|0}} as ''false'', {{math|1}} as ''true'', &and;{{math|1=∧}} as ''and'', &or;{{math|1=∨}} as ''or'', and &not;{{math|¬}} 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 [[logical equivalence|logically equivalent]].

:* The two-element Boolean algebra is also used for circuit design in [[electrical engineering]];{{refn|group=note|Strictly, 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 [[bit]] in a [[digital circuit]], typically high and low [[voltage]]. 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-outputinput–output behavior. Furthermore, every possible input-outputinput–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 always be checked by a trivial [[brute force search|brute force]] algorithm for small numbers of variables). This can for example be used to show that the following laws (''[[Consensus theoremstheorem]]s'') are generally valid in all Boolean algebras:

:** {{math|1=(''a'' &or;∨ ''b'') &and;∧ (&not;¬''a'' &or;∨ ''c'') &and;∧ (''b'' &or;∨ ''c'') &equiv;≡ (''a'' &or;∨ ''b'') &and;∧ (&not;¬''a'' &or;∨ ''c'')}}

:** {{math|1=(''a'' &and;∧ ''b'') &or;∨ (&not;¬''a'' &and;∧ ''c'') &or;∨ (''b'' &and;∧ ''c'') &equiv;≡ (''a'' &and;∧ ''b'') &or;∨ (&not;¬''a'' &and;∧ ''c'')}}

* The [[power set]] (set of all subsets) of any given nonempty set {{math|''S''}} forms a Boolean algebra, an [[algebra of sets]], with the two operations {{math|1=∨ := ∪}} (union) and {{math|1=∧ := ∩}} (intersection). The smallest element 0 is the [[empty set]] and the largest element {{math|1}} is the set {{math|''S''}} itself.

* Starting with the [[propositional calculus]] with &kappa; sentence symbols, form the [[Lindenbaum-Tarski algebra|Lindenbaum algebra]] (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the [[free Boolean algebra]] on &kappa; generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to {0,1}.

:* After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the [[power set]] of two atoms:

* The [[power set]] (set of all subsets) of any given set ''S'' forms a Boolean algebra with the two operations &or; := &cup; (union) and &and; := &cap; (intersection). The smallest element 0 is the [[empty set]] and the largest element 1 is the set ''S'' itself.

{|

|-

* The set of all subsets of ''S'' that are either finite or [[cofinite]] is a Boolean algebra.

| width="70" |

|

* For any [[natural number]] ''n'', the set of all positive [[divisor]]s of ''n'' forms a [[distributive lattice]] if we write ''a'' &le; ''b'' for ''a'' | ''b''. This lattice is a Boolean algebra if and only if ''n'' is [[square-free integer|square-free]]. The smallest element 0 of this Boolean algebra is the natural number 1; the largest element 1 of this Boolean algebra is the natural number ''n''.

{| class="wikitable"

|-

* Other examples of Boolean algebras arise from [[topology|topological spaces]]: if ''X'' is a topological space, then the collection of all subsets of ''X'' which are both open and closed forms a Boolean algebra with the operations &or; := &cup; (union) and &and; := &cap; (intersection).

! {{math|1=∧}} || {{math|1=0}} || {{math|1=a}} || {{math|1=b}} || {{math|1=1}}

|-

* If ''R'' is an arbitrary [[mathematical ring|ring]] and we define the set of ''central idempotents'' by <br> ''A'' = { ''e'' &isin; ''R'' : ''e''² = ''e'', ''ex'' = ''xe'', &forall;''x'' &isin; ''R'' } <br> then the set ''A'' becomes a Boolean algebra with the operations ''e'' &or; ''f'' := ''e'' + ''f'' &minus; ''ef'' and ''e'' &and; ''f'' := ''ef''.

! {{math|1=0}}

| {{math|1=0}} || {{math|1=0}} || {{math|1=0}} || {{math|1=0}}

== Order theoretic properties ==

|-

! {{math|1=a}}

[[Image:Hasse diagram of powerset of 3.png|right|thumb|250px|Boolean lattice of subsets]]

| {{math|1=0}} || {{math|1=a}} || {{math|1=0}} || {{math|1=a}}

Like any lattice, a Boolean algebra (''A'', <math>\land</math>, <math>\lor</math>) gives rise to a [[partially ordered set]] (''A'', &le;) by defining

|-

: ''a'' &le; ''b'' [[iff]] ''a'' = ''a'' <math>\land</math> ''b''

! {{math|1=b}}

(which is also equivalent to ''b'' = ''a'' <math>\lor</math> ''b'').

| {{math|1=0}} || {{math|1=0}} || {{math|1=b}} || {{math|1=b}}

|-

In fact one can also define a Boolean algebra to be a distributive lattice (''A'', &le;) (considered as a partially ordered set) with least element 0 and greatest element 1, within which every element ''x'' has a complement ¬''x'' such that

! {{math|1=1}}

| {{math|1=0}} || {{math|1=a}} || {{math|1=b}} || {{math|1=1}}

|}

|

| width="30" |

|

{| class="wikitable"

|-

! {{math|1=∨}} || {{math|1=0}} || {{math|1=a}} || {{math|1=b}} || {{math|1=1}}

|-

! {{math|1=0}}

| {{math|1=0}} || {{math|1=a}} || {{math|1=b}} || {{math|1=1}}

|-

! {{math|1=a}}

| {{math|1=a}} || {{math|1=a}} || {{math|1=1}} || {{math|1=1}}

|-

! {{math|1=b}}

| {{math|1=b}} || {{math|1=1}} || {{math|1=b}} || {{math|1=1}}

|-

! {{math|1=1}}

| {{math|1=1}} || {{math|1=1}} || {{math|1=1}} || {{math|1=1}}

|}

|

| width="40" |

|

{| class="wikitable"

|-

! {{math|1=''x''}} || {{math|1=0}} || {{math|1=a}} || {{math|1=b}} || {{math|1=1}}

|-

! {{math|1=¬''x''}}

| {{math|1=1}} || {{math|1=b}} || {{math|1=a}} || {{math|1=0}}

|}

|}

* The set {{mvar|A}} of all subsets of {{mvar|S}} that are either finite or [[cofinite]] is a Boolean algebra and an [[algebra of sets]] called the [[finite–cofinite algebra]]. If {{mvar|S}} is infinite then the set of all cofinite subsets of {{mvar|S}}, which is called the [[Fréchet filter]], is a free [[ultrafilter]] on {{mvar|A}}. However, the Fréchet filter is not an ultrafilter on the power set of {{mvar|S}}.

: ''x'' <math>\land</math> ¬''x'' = 0 and ''x'' <math>\lor</math> ¬''x'' = 1

* Starting with the [[propositional calculus]] with {{math|κ}} sentence symbols, form the [[Lindenbaum–Tarski algebra|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 {{math|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 [[linearly ordered]] set {{math|''L''}} with a least element, the interval algebra is the smallest Boolean algebra of subsets of {{math|''L''}} containing all of the half-open intervals {{math|[''a'', ''b'')}} such that {{math|''a''}} is in {{math|''L''}} and {{math|''b''}} is either in {{math|''L''}} or equal to {{math|∞}}. Interval algebras are useful in the study of [[Lindenbaum–Tarski algebra]]s; every [[countable]] Boolean algebra is isomorphic to an interval algebra.

[[File:Lattice T 30.svg|thumb|x150px|[[Hasse diagram]] of the Boolean algebra of divisors of 30.]]

Here <math>\land</math> and <math>\lor</math> are used to denote the [[infimum]] (meet) and [[supremum]] (join) of two elements. Again, if complements in the above sense exist, then they are uniquely determined.

* For any [[natural number]] {{math|''n''}}, the set of all positive [[divisor]]s of {{math|''n''}}, defining {{math|''a'' ≤ ''b''}} if {{math|''a''}} [[divides]] {{math|''b''}}, forms a [[distributive lattice]]. This lattice is a Boolean algebra if and only if {{math|''n''}} is [[square-free integer|square-free]]. The bottom and the top elements of this Boolean algebra are the natural numbers {{math|1}} and {{math|''n''}}, respectively. The complement of {{math|''a''}} is given by {{math|''n''/''a''}}. The meet and the join of {{math|''a''}} and {{math|''b''}} are given by the [[greatest common divisor]] ({{math|gcd}}) and the [[least common multiple]] ({{math|lcm}}) of {{math|''a''}} and {{math|''b''}}, respectively. The ring addition {{math|''a'' + ''b''}} is given by {{math|lcm(''a'', ''b'') / gcd(''a'', ''b'')}}. The picture shows an example for {{math|1=''n'' = 30}}. As a counter-example, considering the non-square-free {{math|1=''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 [[topology|topological spaces]]: if {{math|''X''}} is a topological space, then the collection of all subsets of {{math|''X''}} that are [[Clopen set|both open and closed]] forms a Boolean algebra with the operations {{math|1=∨ := ∪}} (union) and {{math|1=∧ := ∩}} (intersection).

The algebraic and the order theoretic perspective can usually can be used interchangeably and both are of great use to import results and concepts from both [[universal algebra]] and [[order theory]]. In many practical examples an ordering relation, conjunction, disjunction, and negation are all naturally available, so that it is straightforward to exploit this relationship.

* If {{mvar|R}} is an arbitrary ring then its set of ''[[central idempotent]]s'', which is the set

<math display=block>A = \left\{e \in R : e^2 = e \text{ and } ex = xe \; \text{ for all } \; x \in R\right\},</math>

== Principle of duality ==

becomes a Boolean algebra when its operations are defined by {{math|1=''e'' ∨ ''f'' := ''e'' + ''f'' − ''ef''}} and {{math|1=''e'' ∧ ''f'' := ''ef''}}.

One can also apply general insights from [[duality (order theory)|duality in order theory]] to Boolean algebras. Especially, the order dual of every Boolean algebra, or, equivalently, the algebra obtained by exchanging <math>\land</math> and <math>\lor</math>, is also a Boolean algebra. In general, any law valid for Boolean algebras can be transformed into another valid, dual law by exchanging 0 with 1, <math>\land</math> with <math>\lor</math>, and &le; with &ge;.

== Other notation ==

The operators of Boolean algebra may be represented in various ways. Often they

are simply written as AND, OR and NOT. In describing circuits, NAND (NOT AND),

NOR (NOT OR) and XOR (eXclusive OR) may also be used. [[Mathematician]]s, [[engineer]]s, and [[programmer]]s often use + for OR and · for AND (since in some ways those operations are analogous to addition and multiplication in other [[algebraic structure]]s and this notation makes it very easy to get [[sum of products form]] for people who are familiar with normal algebra) and represent NOT by a line drawn above the expression being negated. Sometimes, the symbol ~ is used for NOT.

Here we use another common notation with "meet" <math>\land</math> for AND, "join" <math>\lor</math> for OR, and &not; for NOT.

== Homomorphisms and isomorphisms ==

<!-- "Boolean homomorphism" redirects here -->

A ''[[homomorphism]]'' between thetwo Boolean algebras {{math|''A''}} and {{math|''B''}} is a [[function (mathematics)|function]] {{math|''f'' : ''A'' &rarr;→ ''B''}} such that for all {{math|''a''}}, {{math|''b''}} in {{math|''A''}}:

: {{math|1=''f''(''a'' ∨ ''b'') = ''f''(''a'') ∨ ''f''(''b'')}},

: {{math|1=''f''(''a'' ∧ ''b'') = ''f''(''a'') ∧ ''f''(''b'')}},

: {{math|1=''f''(0) = 0}},

: {{math|1=''f''(1) = 1}}.

It then follows that {{math|1=''f''(¬''a'') = ¬''f''(''a'')}} for all {{math|''a''}} in {{math|''A''}}. The [[class (set theory)|class]] of all Boolean algebras, together with this notion of morphism, forms a [[full subcategory]] of the [[category theory|category]] of lattices.

: ''f''(''a'' <math>\lor</math> ''b'') = ''f''(''a'') <math>\lor</math> ''f''(''b'')

<!--The constant function with ''f''(''a'') = 1 for all ''a'' in ''A'' satisfies the first, second, and fourth conditions but not the third (unless ''B'' is the degenerate singleton Boolean algebra with 0 = 1), so it is not a Boolean algebra homomorphism.-->

: ''f''(''a'' <math>\land</math> ''b'') = ''f''(''a'') <math>\land</math> ''f''(''b'')

: ''f''(0) = 0

: ''f''(1) = 1

An ''isomorphism'' between two Boolean algebras {{math|''A''}} and {{math|''B''}} is a homomorphism {{math|''f'' : ''A'' → ''B''}} with an inverse homomorphism, that is, a homomorphism {{math|''g'' : ''B'' → ''A''}} such that the [[function composition|composition]] {{math|''g'' ∘ ''f'' : ''A'' → ''A''}} is the [[identity function]] on {{math|''A''}}, and the composition {{math|''f'' ∘ ''g'' : ''B'' → ''B''}} is the identity function on {{math|''B''}}. A homomorphism of Boolean algebras is an isomorphism if and only if it is [[bijection|bijective]].

It then follows that ''f''(¬''a'') = ¬''f''(''a'') for all ''a'' in ''A'' as well. The [[class (set theory)|class]] of all Boolean algebras, together with this notion of morphism, forms a [[category theory|category]]. An ''isomorphism'' from ''A'' to ''B'' is a homomorphism from ''A'' to ''B'' which is [[bijective]]. The inverse of an isomorphism is also an isomorphism, and we call the two Boolean algebras ''A'' and ''B'' ''isomorphic''. From the standpoint of Boolean algebra theory, they cannot be distinguished; they differ only in the notation of their elements.

== Boolean rings, ideals and filters ==

{{Main|Boolean ring}}

Every Boolean algebra {{math|1=(''A'', ∧, ∨)}} gives rise to a [[ring (algebra)|ring]] {{math|(''A'', +, ·)}} by defining {{math|1=''a'' + ''b'' := (''a'' ∧ ¬''b'') ∨ (''b'' ∧ ¬''a'') = (''a'' ∨ ''b'') ∧ ¬(''a'' ∧ ''b'')}} (this operation is called [[symmetric difference]] in the case of sets and [[Truth table#Exclusive disjunction|XOR]] in the case of logic) and {{math|1=''a'' · ''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 the {{math|1}} of the Boolean algebra. This ring has the property that {{math|1=''a'' · ''a'' = ''a''}} for all {{math|''a''}} in {{math|''A''}}; rings with this property are called [[Boolean ring]]s.

Conversely, if a Boolean ring {{math|''A''}} is given, we can turn it into a Boolean algebra by defining {{math|1=''x'' ∨ ''y'' := ''x'' + ''y'' + (''x'' · ''y'')}} and {{math|1=''x'' ∧ ''y'' := ''x'' · ''y''}}.{{sfn|Stone|1936}}{{sfn|Hsiang|1985|p=260}}

Every Boolean algebra (''A'', <math>\land</math>, <math>\lor</math>) gives rise to a [[ring (algebra)|ring]] (''A'', +, *) by defining ''a'' + ''b'' = (''a'' <math>\land</math> ¬''b'') <math>\lor</math> (''b'' <math>\land</math> ¬''a'') (this operation is called "symmetric difference" in the case of sets and [[Truth table|XOR]] in the case of logic) and ''a'' * ''b'' = ''a'' <math>\land</math> ''b''. 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 [[Boolean ring]]s.

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 {{math|''f'' : ''A'' → ''B''}} is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The [[category theory|categories]] of Boolean rings and Boolean algebras are [[equivalence of categories|equivalent]];{{sfn|Cohn|2003|p=[https://books.google.com/books?id=VESm0MJOiDQC&pg=PA81 81]}} in fact the categories are [[Isomorphism of categories|isomorphic]].

Hsiang (1985) gave a [[Abstract rewriting system|rule-based algorithm]] to [[Word problem (mathematics)|check]] whether two arbitrary expressions denote the same value in every Boolean ring.

Conversely, if a Boolean ring ''A'' is given, we can turn it into a Boolean algebra by defining ''x'' <math>\lor</math> ''y'' = ''x'' + ''y'' &minus; ''xy'' and ''x'' <math>\land</math> ''y'' = ''xy''. Since these two operations 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'' &rarr; ''B'' is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The [[category theory|categories]] of Boolean rings and Boolean algebras are equivalent.

<!---probably too much details(?):---

Hsiang (1985) gave a [[Confluence (abstract rewriting)|confluent]] and [[Abstract rewriting system#Termination and convergence|terminating]] [[Abstract rewriting system|rewrite system]] for Boolean rings, thus solving their [[Word problem (mathematics)|word problem]]: to check whether two arbitrary expressions ''s'' and ''t'' denote the same value in every Boolean ring, apply rewrite rules to ''s'' as long as possible, resulting in an expression ''s''_n, obtain ''t''_n from ''t'' in a similar way, and check whether ''s''_n and ''t''_n are literally identical, except for different parenthezation and order of operands of "+" or "·".

--->

More generally, Boudet, [[Jean-Pierre Jouannaud|Jouannaud]], and Schmidt-Schauß (1989) gave an algorithm to [[Unification (computer science)#Particular background knowledge sets E|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'' <math>\lor</math> ''y'' in ''I'' and for all ''a'' in ''A'' we have ''a'' <math>\land</math> ''x'' in ''I''. This notion of ideal coincides with the notion of [[ring ideal]] in the Boolean ring ''A''. An ideal ''I'' of ''A'' is called ''prime'' if ''I'' &ne; ''A'' and if ''a'' <math>\land</math> ''b'' in ''I'' always implies ''a'' in ''I'' or ''b'' in ''I''. An ideal ''I'' of ''A'' is called ''maximal'' if ''I'' &ne; ''A'' and if the only ideal properly containing ''I'' is ''A'' itself. These notions coincide with ring theoretic ones of [[prime ideal]] and [[maximal ideal]] in the Boolean ring ''A''.

{{Main|Ideal (order theory)|Filter (mathematics)}}

An ''ideal'' of the Boolean algebra {{mvar|A}} is a nonempty subset {{mvar|I}} such that for all {{mvar|x}}, {{mvar|y}} in {{mvar|I}} we have {{math|{{var|x}} ∨ {{var|y}}}} in {{mvar|I}} and for all {{mvar|a}} in {{mvar|A}} we have {{math|{{var|a}} ∧ {{var|x}}}} in {{mvar|I}}. This notion of ideal coincides with the notion of [[ring ideal]] in the Boolean ring {{mvar|A}}. An ideal {{mvar|I}} of {{mvar|A}} is called ''prime'' if {{math|{{var|I}} ≠ {{var|A}}}} and if {{math|{{var|a}} ∧ {{var|b}}}} in {{mvar|I}} always implies {{mvar|a}} in {{mvar|I}} or {{mvar|b}} in {{mvar|I}}. Furthermore, for every {{math|{{var|a}} ∈ {{var|A}}}} we have that {{math|{{var|a}} ∧ &minus;{{var|a}} {{=}} 0 ∈ {{var|I}}}}, and then if {{mvar|I}} is prime we have {{math|{{var|a}} ∈ {{var|I}}}} or {{math|&minus;{{var|a}} ∈ {{var|I}}}} for every {{math|{{var|a}} ∈ {{var|A}}}}. An ideal {{mvar|I}} of {{mvar|A}} is called ''maximal'' if {{math|{{var|I}} ≠ {{var|A}}}} and if the only ideal properly containing {{mvar|I}} is {{mvar|A}} itself. For an ideal {{mvar|I}}, if {{math|{{var|a}} ∉ {{var|I}}}} and {{math|&minus;{{var|a}} ∉ {{var|I}}}}, then {{math|{{var|I}} ∪ {{mset|{{var|a}}}}}} or {{math|{{var|I}} ∪ {{mset|&minus;{{var|a}}}}}} is contained in another proper ideal {{mvar|J}}. Hence, such an {{mvar|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 [[prime ideal]] and [[maximal ideal]] in the Boolean ring {{mvar|A}}.

The dual of an ''ideal'' is a ''filter''. A ''filter'' of the Boolean algebra ''{{mvar|A''}} is a nonempty subset ''{{mvar|p''}} such that for all ''{{mvar|x''}}, ''{{mvar|y''}} in ''{{mvar|p''}} we have ''{{math|{{var|x''}} <math>\land</math>∧ ''{{var|y''}}}} in ''{{mvar|p''}} and for all ''{{mvar|a''}} in {{mvar|A}} we have {{math|{{var|a}} ∨ {{var|x}}}} in {{mvar|p}}. The dual of a ''Amaximal'' if(or ''aprime'' <math>\lor</math>) ''xideal'' =in a Boolean algebra is ''a[[ultrafilter]]''. Ultrafilters thencan alternatively be described as [[2-valued morphism]]s from {{mvar|A}} to the two-element Boolean algebra. The statement ''every filter in a Boolean algebra can be extended to an ultrafilter'' inis called the ''p[[Boolean prime ideal theorem#The ultrafilter lemma|ultrafilter lemma]]'' and cannot be proven in [[Zermelo–Fraenkel set theory]] (ZF), if [[Zermelo–Fraenkel set theory|ZF]] is [[consistent]]. Within ZF, the ultrafilter lemma is strictly weaker than the [[axiom 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.

== Representations ==

== Representing Boolean algebras ==

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]].

[[Marshall H. Stone|Stone's]] celebrated ''[[Stone's representation theorem for Boolean algebras|representation theorem for Boolean algebras]]'' states that ''every'' Boolean algebra {{math|''A''}} is isomorphic to the Boolean algebra of all [[closed-openclopen set]] setss in some ([[compact space|compact]] [[totally disconnected]] [[Hausdorff space|Hausdorff]]) topological space.

== Axiomatics for Boolean algebras ==

{| align="right" class="wikitable collapsible collapsed" style="text-align:left"

In 1933, the American mathematician [[Edward Vermilye Huntington]] (1874-1952) showed the following axiomatization for Boolean algebra:

! colspan="2" | '''Proven properties'''

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''UId₁''' !! !! colspan="2" | If ''x'' ∨ ''o'' = ''x'' for all ''x'', then ''o'' = 0

|-

| Proof: || || colspan="2" | If ''x'' ∨ ''o'' = ''x'', then

|-

| || || 0

|-

| || = || 0 ∨ ''o'' || by assumption

|-

| || = || ''o'' ∨ 0 || by '''Cmm₁'''

|-

| || = || ''o'' || by '''Idn₁'''

|}

| '''UId₂''' &nbsp; [dual] &nbsp; If ''x'' ∧ ''i'' = ''x'' for all ''x'', then ''i'' = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''Idm₁''' !! !! ''x'' ∨ ''x'' = ''x''

|-

| Proof: || || ''x'' ∨ ''x''

|-

| || = || (''x'' ∨ ''x'') ∧ 1 || by '''Idn₂'''

|-

| || = || (''x'' ∨ ''x'') ∧ (''x'' ∨ ¬''x'') || by '''Cpl₁'''

|-

| || = || ''x'' ∨ (''x'' ∧ ¬''x'') || by '''Dst₁'''

|-

| || = || ''x'' ∨ 0 || by '''Cpl₂'''

|-

| || = || ''x'' || by '''Idn₁'''

|}

| '''Idm₂''' &nbsp; [dual] &nbsp; ''x'' ∧ ''x'' = ''x''

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''Bnd₁''' !! !! ''x'' ∨ 1 = 1

|-

| Proof: || || ''x'' ∨ 1

|-

| || = || (''x'' ∨ 1) ∧ 1 || by '''Idn₂'''

|-

| || = || 1 ∧ (''x'' ∨ 1) || by '''Cmm₂'''

|-

| || = || (''x'' ∨ ¬''x'') ∧ (''x'' ∨ 1) || by '''Cpl₁'''

|-

| || = || ''x'' ∨ (¬''x'' ∧ 1) || by '''Dst₁'''

|-

| || = || ''x'' ∨ ¬''x'' || by '''Idn₂'''

|-

| || = || 1 || by '''Cpl₁'''

|}

| '''Bnd₂''' &nbsp; [dual] &nbsp; ''x'' ∧ 0 = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''Abs₁''' !! !! ''x'' ∨ (''x'' ∧ ''y'') = ''x''

|-

| Proof: || || ''x'' ∨ (''x'' ∧ ''y'')

|-

| || = || (''x'' ∧ 1) ∨ (''x'' ∧ ''y'') || by '''Idn₂'''

|-

| || = || ''x'' ∧ (1 ∨ ''y'') || by '''Dst₂'''

|-

| || = || ''x'' ∧ (''y'' ∨ 1) || by '''Cmm₁'''

|-

| || = || ''x'' ∧ 1 || by '''Bnd₁'''

|-

| || = || ''x'' || by '''Idn₂'''

|}

| '''Abs₂''' &nbsp; [dual] &nbsp; ''x'' ∧ (''x'' ∨ ''y'') = ''x''

|- valign="top"

| colspan="2" |

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''UNg''' !! !! colspan="2" | If ''x'' ∨ ''x''_n = 1 and ''x'' ∧ ''x''_n = 0, then ''x''_n = ¬''x''

|-

| Proof: || || colspan="2" | If ''x'' ∨ ''x''_n = 1 and ''x'' ∧ ''x''_n = 0, then

|-

| || ||''x''_n

|-

| || = || ''x''_n ∧ 1 || by '''Idn₂'''

|-

| || = || ''x''_n ∧ (''x'' ∨ ¬''x'') || by '''Cpl₁'''

|-

| || = || (''x''_n ∧ ''x'') ∨ (''x''_n ∧ ¬''x'') || by '''Dst₂'''

|-

| || = || (''x'' ∧ ''x''_n) ∨ (¬''x'' ∧ ''x''_n) || by '''Cmm₂'''

|-

| || = || 0 ∨ (¬''x'' ∧ ''x''_n) || by assumption

|-

| || = || (''x'' ∧ ¬''x'') ∨ (¬''x'' ∧ ''x''_n) || by '''Cpl₂'''

|-

| || = || (¬''x'' ∧ ''x'') ∨ (¬''x'' ∧ ''x''_n) || by '''Cmm₂'''

|-

| || = || ¬''x'' ∧ (''x'' ∨ ''x''_n) || by '''Dst₂'''

|-

| || = || ¬''x'' ∧ 1 || by assumption

|-

| || = || ¬''x'' || by '''Idn₂'''

|}

|- valign="top"

| colspan="2" |

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''DNg''' !! !! ¬¬''x'' = ''x''

|-

| Proof: || || ¬''x'' ∨ ''x'' = ''x'' ∨ ¬''x'' = 1 || by '''Cmm₁''', '''Cpl₁'''

|-

| || and || ¬''x'' ∧ ''x'' = ''x'' ∧ ¬''x'' = 0 || by '''Cmm₂''', '''Cpl₂'''

|-

| || hence || ''x'' = ¬¬''x'' || by '''UNg'''

|}

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''A₁''' !! !! ''x'' ∨ (¬''x'' ∨ ''y'') = 1

|-

| Proof: || || ''x'' ∨ (¬''x'' ∨ ''y'')

|-

| || = || (''x'' ∨ (¬''x'' ∨ ''y'')) ∧ 1 || by '''Idn₂'''

|-

| || = || 1 ∧ (''x'' ∨ (¬''x'' ∨ ''y'')) || by '''Cmm₂'''

|-

| || = || (''x'' ∨ ¬''x'') ∧ (''x'' ∨ (¬''x'' ∨ ''y'')) || by '''Cpl₁'''

|-

| || = || ''x'' ∨ (¬''x'' ∧ (¬''x'' ∨ ''y'')) || by '''Dst₁'''

|-

| || = || ''x'' ∨ ¬''x'' || by '''Abs₂'''

|-

| || = || 1 || by '''Cpl₁'''

|}

| '''A₂''' &nbsp; [dual] &nbsp; ''x'' ∧ (¬''x'' ∧ ''y'') = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''B₁''' !! !! (''x'' ∨ ''y'') ∨ (¬''x'' ∧ ¬''y'') = 1

|-

| Proof: || || (''x'' ∨ ''y'') ∨ (¬''x'' ∧ ¬''y'')

|-

| || = || ((''x'' ∨ ''y'') ∨ ¬''x'') ∧ ((''x'' ∨ ''y'') ∨ ¬''y'') || by '''Dst₁'''

|-

| || = || (¬''x'' ∨ (''x'' ∨ ''y'')) ∧ (¬''y'' ∨ (''y'' ∨ ''x'')) || by '''Cmm₁'''

|-

| || = || (¬''x'' ∨ (¬¬''x'' ∨ ''y'')) ∧ (¬''y'' ∨ (¬¬''y'' ∨ ''x'')) || by '''DNg'''

|-

| || = || 1 ∧ 1 || by '''A₁'''

|-

| || = || 1 || by '''Idn₂'''

|}

| '''B₂''' &nbsp; [dual] &nbsp; (''x'' ∧ ''y'') ∧ (¬''x'' ∨ ¬''y'') = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''C₁''' !! !! (''x'' ∨ ''y'') ∧ (¬''x'' ∧ ¬''y'') = 0

|-

| Proof: || || (''x'' ∨ ''y'') ∧ (¬''x'' ∧ ¬''y'')

|-

| || = || (¬''x'' ∧ ¬''y'') ∧ (''x'' ∨ ''y'') || by '''Cmm₂'''

|-

| || = || ((¬''x'' ∧ ¬''y'') ∧ ''x'') ∨ ((¬''x'' ∧ ¬''y'') ∧ ''y'') || by '''Dst₂'''

|-

| || = || (''x'' ∧ (¬''x'' ∧ ¬''y'')) ∨ (''y'' ∧ (¬''y'' ∧ ¬''x'')) || by '''Cmm₂'''

|-

| || = || 0 ∨ 0 || by '''A₂'''

|-

| || = || 0 || by '''Idn₁'''

|}

| '''C₂''' &nbsp; [dual] &nbsp; (''x'' ∧ ''y'') ∨ (¬''x'' ∨ ¬''y'') = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''DMg₁''' !! !! ¬(''x'' ∨ ''y'') = ¬''x'' ∧ ¬''y''

|-

| Proof: || || by '''B₁''', '''C₁''', and '''UNg'''

|}

| '''DMg₂''' &nbsp; [dual] &nbsp; ¬(''x'' ∧ ''y'') = ¬''x'' ∨ ¬''y''

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''D₁''' !! !! (''x''∨(''y''∨''z'')) ∨ ¬''x'' = 1

|-

| Proof: || || (''x'' ∨ (''y'' ∨ ''z'')) ∨ ¬''x''

|-

| || = || ¬''x'' ∨ (''x'' ∨ (''y'' ∨ ''z'')) || by '''Cmm₁'''

|-

| || = || ¬''x'' ∨ (¬¬''x'' ∨ (''y'' ∨ ''z'')) || by '''DNg'''

|-

| || = || 1 || by '''A₁'''

|}

| '''D₂''' &nbsp; [dual] &nbsp; (''x''∧(''y''∧''z'')) ∧ ¬''x'' = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''E₁''' !! !! ''y'' ∧ (''x''∨(''y''∨''z'')) = ''y''

|-

| Proof: || || ''y'' ∧ (''x'' ∨ (''y'' ∨ ''z''))

|-

| || = || (''y'' ∧ ''x'') ∨ (''y'' ∧ (''y'' ∨ ''z'')) || by '''Dst₂'''

|-

| || = || (''y'' ∧ ''x'') ∨ ''y'' || by '''Abs₂'''

|-

| || = || ''y'' ∨ (''y'' ∧ ''x'') || by '''Cmm₁'''

|-

| || = || ''y'' || by '''Abs₁'''

|}

| '''E₂''' &nbsp; [dual] &nbsp; ''y'' ∨ (''x''∧(''y''∧''z'')) = ''y''

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''F₁''' !! !! (''x''∨(''y''∨''z'')) ∨ ¬''y'' = 1

|-

| Proof: || || (''x'' ∨ (''y'' ∨ ''z'')) ∨ ¬''y''

|-

| || = || ¬''y'' ∨ (''x'' ∨ (''y'' ∨ ''z'')) || by '''Cmm₁'''

|-

| || = || (¬''y'' ∨ (''x'' ∨ (''y'' ∨ ''z''))) ∧ 1 || by '''Idn₂'''

|-

| || = || 1 ∧ (¬''y'' ∨ (''x'' ∨ (''y'' ∨ ''z''))) || by '''Cmm₂'''

|-

| || = || (''y'' ∨ ¬''y'') ∧ (¬''y'' ∨ (''x'' ∨ (''y'' ∨ ''z''))) || by '''Cpl₁'''

|-

| || = || (¬''y'' ∨ ''y'') ∧ (¬''y'' ∨ (''x'' ∨ (''y'' ∨ ''z''))) || by '''Cmm₁'''

|-

| || = || ¬''y'' ∨ (''y'' ∧ (''x'' ∨ (''y'' ∨ ''z''))) || by '''Dst₁'''

|-

| || = || ¬''y'' ∨ ''y'' || by '''E₁'''

|-

| || = || ''y'' ∨ ¬''y'' || by '''Cmm₁'''

|-

| || = || 1 || by '''Cpl₁'''

|}

| '''F₂''' &nbsp; [dual] &nbsp; (''x''∧(''y''∧''z'')) ∧ ¬''y'' = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''G₁''' !! !! (''x''∨(''y''∨''z'')) ∨ ¬''z'' = 1

|-

| Proof: || || (''x'' ∨ (''y'' ∨ ''z'')) ∨ ¬''z''

|-

| || = || (''x'' ∨ (''z'' ∨ ''y'')) ∨ ¬''z'' || by '''Cmm₁'''

|-

| || = || 1 || by '''F₁'''

|}

| '''G₂''' &nbsp; [dual] &nbsp; (''x''∧(''y''∧''z'')) ∧ ¬''z'' = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''H₁''' !! !! ¬((''x''∨''y'')∨''z'') ∧ ''x'' = 0

|-

| Proof: || || ¬((''x'' ∨ ''y'') ∨ ''z'') ∧ ''x''

|-

| || = || (¬(''x'' ∨ ''y'') ∧ ¬''z'') ∧ ''x'' || by '''DMg₁'''

|-

| || = || ((¬''x'' ∧ ¬''y'') ∧ ¬''z'') ∧ ''x'' || by '''DMg₁'''

|-

| || = || ''x'' ∧ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'') || by '''Cmm₂'''

|-

| || = || (''x'' ∧ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'')) ∨ 0 || by '''Idn₁'''

|-

| || = || 0 ∨ (''x'' ∧ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'')) || by '''Cmm₁'''

|-

| || = || (''x'' ∧ ¬''x'') ∨ (''x'' ∧ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'')) || by '''Cpl₂'''

|-

| || = || ''x'' ∧ (¬''x'' ∨ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'')) || by '''Dst₂'''

|-

| || = || ''x'' ∧ (¬''x'' ∨ (¬''z'' ∧ (¬''x'' ∧ ¬''y''))) || by '''Cmm₂'''

|-

| || = || ''x'' ∧ ¬''x'' || by '''E₂'''

|-

| || = || 0 || by '''Cpl₂'''

|}

| '''H₂''' &nbsp; [dual] &nbsp; ¬((''x''∧''y'')∧''z'') ∨ ''x'' = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''I₁''' !! !! ¬((''x''∨''y'')∨''z'') ∧ ''y'' = 0

|-

| Proof: || || ¬((''x'' ∨ ''y'') ∨ ''z'') ∧ ''y''

|-

| || = || ¬((''y'' ∨ ''x'') ∨ ''z'') ∧ ''y'' || by '''Cmm₁'''

|-

| || = || 0 || by '''H₁'''

|}

| '''I₂''' &nbsp; [dual] &nbsp; ¬((''x''∧''y'')∧''z'') ∨ ''y'' = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''J₁''' !! !! ¬((''x''∨''y'')∨''z'') ∧ ''z'' = 0

|-

| Proof: || || ¬((''x'' ∨ ''y'') ∨ ''z'') ∧ ''z''

|-

| || = || (¬(''x'' ∨ ''y'') ∧ ¬''z'') ∧ ''z'' || by '''DMg₁'''

|-

| || = || ''z'' ∧ (¬(''x'' ∨ ''y'') ∧ ¬''z'') || by '''Cmm₂'''

|-

| || = || ''z'' ∧ ( ¬''z'' ∧ ¬(''x'' ∨ ''y'')) || by '''Cmm₂'''

|-

| || = || 0 || by '''A₂'''

|}

| '''J₂''' &nbsp; [dual] &nbsp; ¬((''x''∧''y'')∧''z'') ∨ ''z'' = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''K₁''' !! !! (''x'' ∨ (''y'' ∨ ''z'')) ∨ ¬((''x'' ∨ ''y'') ∨ ''z'') = 1

|-

| Proof: || || (''x''∨(''y''∨''z'')) ∨ ¬((''x'' ∨ ''y'') ∨ ''z'')

|-

| || = || (''x''∨(''y''∨''z'')) ∨ (¬(''x'' ∨ ''y'') ∧ ¬''z'') || by '''DMg₁'''

|-

| || = || (''x''∨(''y''∨''z'')) ∨ ((¬''x'' ∧ ¬''y'') ∧ ¬''z'') || by '''DMg₁'''

|-

| || = || ((''x''∨(''y''∨''z'')) ∨ (¬''x'' ∧ ¬''y'')) ∧ ((''x''∨(''y''∨''z'')) ∨ ¬''z'')|| by '''Dst₁'''

|-

| || = || (((''x''∨(''y''∨''z'')) ∨ ¬''x'') ∧ ((''x''∨(''y''∨''z'')) ∨ ¬''y'')) ∧ ((''x''∨(''y''∨''z'')) ∨ ¬''z'')|| by '''Dst₁'''

|-

| || = || (1 ∧ 1) ∧ 1 || by '''D₁''','''F₁''','''G₁'''

|-

| || = || 1 || by '''Idn₂'''

|}

| '''K₂''' &nbsp; [dual] &nbsp; (''x'' ∧ (''y'' ∧ ''z'')) ∧ ¬((''x'' ∧ ''y'') ∧ ''z'') = 0

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''L₁''' !! !! (''x'' ∨ (''y'' ∨ ''z'')) ∧ ¬((''x'' ∨ ''y'') ∨ ''z'') = 0

|-

| Proof: || || (''x'' ∨ (''y'' ∨ ''z'')) ∧ ¬((''x'' ∨ ''y'') ∨ ''z'')

|-

| || = || ¬((''x''∨''y'')∨''z'') ∧ (''x'' ∨ (''y'' ∨ ''z'')) || by '''Cmm₂'''

|-

| || = || (¬((''x''∨''y'')∨''z'') ∧ ''x'') ∨ (¬((''x''∨''y'')∨''z'') ∧ (''y'' ∨ ''z'')) || by '''Dst₂'''

|-

| || = || (¬((''x''∨''y'')∨''z'') ∧ ''x'') ∨ ((¬((''x''∨''y'')∨''z'') ∧ ''y'') ∨ (¬((''x''∨''y'')∨''z'') ∧ ''z'')) || by '''Dst₂'''

|-

| || = || 0 ∨ (0 ∨ 0) || by '''H₁''','''I₁''','''J₁'''

|-

| || = || 0 || by '''Idn₁'''

|}

| '''L₂''' &nbsp; [dual] &nbsp; (''x'' ∧ (''y'' ∧ ''z'')) ∨ ¬((''x'' ∧ ''y'') ∧ ''z'') = 1

|- valign="top"

|

{| align="left" class="collapsible collapsed" style="text-align:left"

! '''Ass₁''' !! !! ''x'' ∨ (''y'' ∨ ''z'') = (''x'' ∨ ''y'') ∨ ''z''

|-

| Proof: || || by '''K₁''', '''L₁''', '''UNg''', '''DNg'''

|}

| '''Ass₂''' &nbsp; [dual] &nbsp; ''x'' ∧ (''y'' ∧ ''z'') = (''x'' ∧ ''y'') ∧ ''z''

|-

| colspan="2" |

{| align="left" class="collapsible" style="text-align:left"

|-

! colspan="2" | Abbreviations

|-

| '''UId''' || Unique Identity

|-

| '''Idm''' || [[Idempotence]]

|-

| '''Bnd''' || [[Bounded lattice|Boundaries]]

|-

| '''Abs''' || [[Absorption law]]

|-

| '''UNg''' || Unique Negation

|-

| '''DNg''' || [[Double negation]]

|-

| '''DMg''' || [[De Morgan's Law]]

|-

| '''Ass''' || [[Associativity]]

|}

|}

{| align="right" class="wikitable collapsible collapsed" style="text-align:left"

! colspan="4"| '''Huntington 1904 Boolean algebra axioms'''

|- valign="top"

| '''Idn₁''' || ''x'' ∨ 0 = ''x''

| '''Idn₂''' || ''x'' ∧ 1 = ''x''

|- valign="top"

| '''Cmm₁''' || ''x'' ∨ ''y'' = ''y'' ∨ ''x''

| '''Cmm₂''' || ''x'' ∧ ''y'' = ''y'' ∧ ''x''

|- valign="top"

| '''Dst₁''' || ''x'' ∨ (''y''∧''z'') = (''x''∨''y'') ∧ (''x''∨''z'')

| '''Dst₂''' || ''x'' ∧ (''y''∨''z'') = (''x''∧''y'') ∨ (''x''∧''z'')

|- valign="top"

| '''Cpl₁''' || ''x'' ∨ ¬''x'' = 1

| '''Cpl₂''' || ''x'' ∧ ¬''x'' = 0

|-

| colspan="4" |

{| align="left" class="collapsible" style="text-align:left"

|-

! colspan="2" | Abbreviations

|-

| '''Idn''' || [[Identity element|Identity]]

|-

| '''Cmm''' || [[Commutativity]]

|-

| '''Dst''' || [[Distributivity]]

|-

| '''Cpl''' || [[Complemented lattice|Complements]]

|}

|}

The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician [[Alfred North Whitehead]] in 1898.{{sfn|Padmanabhan|Rudeanu|2008|p=[https://books.google.com/books?id=JlXSlpmlSv4C&pg=PA73 73]}}{{sfn|Whitehead|1969|p=37}}

# ''Commutativity'': ''x'' + ''y'' = ''y'' + ''x''.

It included the [[#Definition|above axioms]] and additionally {{math|1=''x'' ∨ 1 = 1}} and {{math|1=''x'' ∧ 0 = 0}}.

# ''Associativity'': (''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'').

In 1904, the American mathematician [[Edward V. Huntington]] (1874–1952) gave probably the most parsimonious axiomatization based on {{math|1=∧}}, {{math|1=∨}}, {{math|1=¬}}, even proving the associativity laws (see box).{{sfn|Huntington|1904|pp=292-293}}

# ''Huntington equation'': ''n''(''n''(''x'') + ''y'') + ''n''(''n''(''x'') + ''n''(''y'')) = ''x''.

He also proved that these axioms are [[Independence (mathematical logic)|independent]] of each other.{{sfn|Huntington|1904|p=296}}

In 1933, Huntington set out the following elegant axiomatization for Boolean algebra.{{sfn|Huntington|1933a}} It requires just one binary operation {{math|1=+}} and a [[unary functional symbol]] {{math|1=''n''}}, to be read as 'complement', which satisfy the following laws:

{{olist

|1= ''Commutativity'': {{math|1=''x'' + ''y'' = ''y'' + ''x''}}.

|2= ''Associativity'': {{math|1=(''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'')}}.

|3= ''Huntington equation'': {{math|1=''n''(''n''(''x'') + ''y'') + ''n''(''n''(''x'') + ''n''(''y'')) = ''x''}}.

}}

[[Herbert Robbins]] immediately asked: If the Huntington equation is replaced with its dual, to wit:

{{olist|start=4

|1= ''Robbins Equation'': {{math|1=''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 [[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 computer program [[Equational prover|EQP]] he designed. For a simplification of McCune's proof, see Dahn (1998).

Further work has been done for reducing the number of axioms; see [[Minimal axioms for Boolean algebra]].

The [[unary functional symbol]] ''n'' may be read as 'complement'.

{{clear}}

[[Herbert Robbins]] then posed the following question: Can the Huntington equation be shortened as follows, and is this new equation, together with associativity and commutativity, a basis for Boolean algebra? With this collection of axioms called a ''Robbins algebra'', the question then becomes: Is every Robbins algebra a Boolean algebra?

== Generalizations ==

Axiomatization for Robbins algebra:

{{Algebraic structures|Lattice}}

Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a [[distributive lattice]] {{math|1=''B''}} is a generalized Boolean lattice, if it has a smallest element {{math|1=0}} and for any elements {{math|1=''a''}} and {{math|1=''b''}} in {{math|1=''B''}} such that {{math|1=''a'' ≤ ''b''}}, there exists an element {{math|1=''x''}} such that {{math|1=''a'' ∧ ''x'' = 0}} and {{math|1=''a'' ∨ ''x'' = ''b''}}. Defining {{math|1=''a'' \ ''b''}} as the unique {{math|1=''x''}} such that {{math|1=(''a'' ∧ ''b'') ∨ ''x'' = ''a''}} and {{math|1=(''a'' ∧ ''b'') ∧ ''x'' = 0}}, we say that the structure {{math|(''B'', ∧, ∨, \, 0)}} is a ''generalized Boolean algebra'', while {{math|(''B'', ∨, 0)}} is a ''generalized Boolean [[semilattice]]''. Generalized Boolean lattices are exactly the [[Ideal (order theory)|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 [[quantum logic]] as lattices of [[closed set|closed]] [[linear subspace]]s for [[separable space|separable]] [[Hilbert space]]s.

# ''Commutativity'': ''x'' + ''y'' = ''y'' + ''x''.

# ''Associativity'': (''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'').

# ''Robbins Equation'': ''n''(''n''(''x'' + ''y''') + ''n''(''x'' + ''n''(''y''))) = ''x''.

This question remained open from the 1930s, and became a favorite question of [[Alfred Tarski]] and his students.

In 1996, [[William McCune]] at [[Argonne National Laboratory]], building upon the work of Larry Wos, Steve Winker, and Bob Veroff, answered this long-standing question in the affirmative: Every Robbins algebra is a Boolean algebra. This work was done using McCune's [[automated reasoning program]] [[EQP]].

==See also==

== See also ==

{{div col|content=

* [[List of Boolean algebra topics]]

* [[Boolean ___domain]]

* [[Boolean function]]

* [[Boolean logic]]

* [[Boolean ring]]

* [[Boolean-valued function]]

* [[Canonical form (Boolean algebra)]]

* [[Complete Boolean algebra]]

* [[De Morgan's laws]]

* [[Forcing (mathematics)]]

* [[Free Boolean algebra]]

* [[Heyting algebra]]

* [[Hypercube graph]]

* [[Karnaugh map]]

* ''[[ListLaws of Boolean algebra topicsForm]]''

* [[Logic gate]]

* [[Logical graph]]

* [[:simple:Boolean algebra|Boolean algebra]] (an article about Boolean logic in the Simple English Wikipedia)

* [[Logical matrix]]

* [[Propositional logic]]

* [[Quine–McCluskey algorithm]]

* [[Two-element Boolean algebra]]

* [[Venn diagram]]

* [[Conditional event algebra]]

}}

==Notes==

{{reflist|group=note}}

== References ==

{{reflist|30em}}

===Works cited===

*{{cite book

|first1=B.A. |last1=Davey |author2link = Hilary Priestley|first2=H.A. |last2=Priestley | title=Introduction to Lattices and Order | title-link= Introduction to Lattices and Order

| year=1990

| publisher=Cambridge University Press

| series=Cambridge Mathematical Textbooks}}

*{{citation|title=Basic Algebra: Groups, Rings, and Fields|first=Paul M.|last=Cohn|publisher= Springer |year=2003 |isbn=9781852335878 |pages=51, 70–81 |author-link=Paul Cohn}}

*{{citation

| last1 = Givant

| first1 = Steven

| first2 = Paul

| last2 = Halmos

|author2link = Paul Halmos

| year = 2009

| title = Introduction to Boolean Algebras

| series = [[Undergraduate Texts in Mathematics]] | publisher = [[Springer Science+Business Media|Springer]]

| isbn = 978-0-387-40293-2}}.

*{{citation|title=Boolean Algebra|first=R. L.|last=Goodstein|authorlink = Reuben Goodstein|publisher=Courier Dover Publications|year=2012|isbn=9780486154978|chapter=Chapter 2: The self-dual system of axioms|pages=21ff|chapter-url=https://books.google.com/books?id=0fxW2KiyxWwC&pg=PA21}}

*{{cite journal

| last1=Hsiang

| first1=Jieh

| title=Refutational Theorem Proving Using Term Rewriting Systems

| journal= [[Artificial Intelligence (journal)|Artificial Intelligence]]

| year=1985

| volume=25

| issue=3

| pages=255–300

| url=https://www.researchgate.net/publication/223327412

| doi=10.1016/0004-3702(85)90074-8

}}

*{{cite journal

| author-link=Edward V. Huntington

| first1=Edward V.

| last1=Huntington

| title=Sets of Independent Postulates for the Algebra of Logic

| journal=[[Transactions of the American Mathematical Society]]

| year=1904

| volume=5

| issue=3

| pages=288–309

| jstor=1986459

| doi=10.1090/s0002-9947-1904-1500675-4| url=https://zenodo.org/record/1431563

| doi-access=free

}}

*{{citation

| last1 = Padmanabhan

| first1 = Ranganathan

| last2 = Rudeanu

| first2 = Sergiu

| year = 2008

| title = Axioms for lattices and boolean algebras

| publisher = World Scientific

| isbn = 978-981-283-454-6}}.

*{{cite journal

| author-link=Marshall H. Stone

| first1=Marshall H.

| last1=Stone

| title=The Theory of Representations for Boolean Algebra

| journal=[[Transactions of the American Mathematical Society]]

| year=1936

| volume=40

| pages=37–111

| doi=10.1090/s0002-9947-1936-1501865-8| doi-access=free

}}

*{{cite book |author-link=A.N. Whitehead |first1=A.N. |last1=Whitehead |title=A Treatise on Universal Algebra |year=1969 |publisher=Cambridge University Press |isbn=978-1-4297-0032-0 |orig-year=1898}}

=== General references ===

{{more footnotes needed|date=July 2013}}

*{{citation

| last1 = Brown

| first1 = Stephen

| last2 = Vranesic

| first2 = Zvonko

| year = 2002

| title = Fundamentals of Digital Logic with VHDL Design

| edition = 2nd

| publisher = [[McGraw-Hill|McGraw–Hill]]

| isbn = 978-0-07-249938-4}}. See Section 2.5.

*{{cite journal

|first1=A. |last1=Boudet |first2=J.P. |last2=Jouannaud |first3=M. |last3=Schmidt-Schauß | title=Unification in Boolean Rings and Abelian Groups

| journal=[[Journal of Symbolic Computation]]

| year=1989

| volume=8

|issue=5 | pages=449–477

| doi=10.1016/s0747-7171(89)80054-9|doi-access=free }}

*{{citation

| last1 = Cori

| first1 = Rene

| last2 = Lascar

| first2 = Daniel

| year = 2000

| title = Mathematical Logic: A Course with Exercises

| publisher = [[Oxford University Press]]

| isbn = 978-0-19-850048-3}}. See Chapter 2.

*{{citation

| last = Dahn

| first = B. I.

| year = 1998

| title = Robbins Algebras are Boolean: A Revision of McCune's Computer-Generated Solution of the Robbins Problem

| journal = [[Journal of Algebra]]

| volume = 208

| pages = 526–532

| doi = 10.1006/jabr.1998.7467

| issue = 2| doi-access = free

}}.

*{{citation |author-link=Paul Halmos |last=Halmos |first=Paul |year=1963 |title=Lectures on Boolean Algebras |publisher=Van Nostrand}}.

== External links ==

*{{citation

| author-link1 = Paul Halmos

| last1 = Halmos

| first1 = Paul

| first2 = Steven

| last2 = Givant

| year = 1998

| title = Logic as Algebra

| series = Dolciani Mathematical Expositions

| volume = 21

| publisher = [[Mathematical Association of America]]

| isbn = 978-0-88385-327-6

| url-access = registration

| url = https://archive.org/details/logicasalgebra0000halm

}}.

*{{citation

| author-link = Edward Vermilye Huntington

| last = Huntington

| first = E. V.

| year = 1933a

| title = New sets of independent postulates for the algebra of logic

| journal = [[Transactions of the American Mathematical Society]]

| volume = 35

| pages = 274–304

| doi = 10.2307/1989325

| issue = 1

| publisher = American Mathematical Society

| url = https://www.ams.org/journals/tran/1933-035-01/S0002-9947-1933-1501684-X/S0002-9947-1933-1501684-X.pdf

| jstor = 1989325}}.

*{{citation

| author-link = Edward Vermilye Huntington

| last = Huntington

| first = E. V.

| year = 1933b

| title = Boolean algebra: A correction

| journal = [[Transactions of the American Mathematical Society]]

| volume = 35

| pages = 557–558

| doi = 10.2307/1989783

| issue = 2

| jstor = 1989783}}

*{{citation

| last = Mendelson

| first = Elliott

| year = 1970

| title = Boolean Algebra and Switching Circuits

| series = Schaum's Outline Series in Mathematics

| publisher = [[McGraw-Hill|McGraw–Hill]]

| isbn = 978-0-07-041460-0

| url-access = registration

| url = https://archive.org/details/schaumsoutlineof00mend

}}

*{{citation

| editor1-last = Monk

| editor1-first = J. Donald

| editor2-first = R.

| editor2-last = Bonnet

| year = 1989

| title = Handbook of Boolean Algebras

| publisher = [[Elsevier|North-Holland]]

| isbn = 978-0-444-87291-3

| url-access = registration

| url = https://archive.org/details/handbookofboolea0000unse

}}. 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}})

*{{citation

| author-link = Roman Sikorski

| last = Sikorski

| first = Roman

| year = 1966

| title = Boolean Algebras

| series = Ergebnisse der Mathematik und ihrer Grenzgebiete

| publisher = [[Springer Verlag]]

| ref = Sikorski1966BooleanAlgebras

}}.

*{{citation

| last = Stoll

| first = R. R.

| year = 1979|orig-year=1963

| title = Set Theory and Logic

| publisher = W. H. Freeman

| isbn = 978-0-486-63829-4}}. Reprinted by [[Dover Publications]], 1979.

==External links==

* [http://plato.stanford.edu/entries/boolalg-math/ Article on ''The Mathematics of Boolean Algebra''] at the [[Stanford Encyclopedia of Philosophy]]

{{external links|date=November 2020}}

* {{springer|title=Boolean algebra|id=p/b016920}}

* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/boolalg-math/ The Mathematics of Boolean Algebra]", by J. Donald Monk.

* McCune W., 1997. ''[http://www.cs.unm.edu/~mccune/papers/robbins/ Robbins Algebras Are Boolean]'' JAR 19(3), 263–276

* [http://demonstrations.wolfram.com/BooleanAlgebra/ "Boolean Algebra"] by [[Eric W. Weisstein]], [[Wolfram Demonstrations Project]], 2007.

* Burris, Stanley N.; Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. {{ISBN|3-540-90578-2}}.

* {{MathWorld | urlname=BooleanAlgebra | title=Boolean Algebra}}

{{Order theory}}

[[Category:Boolean algebra|*]]

{{DEFAULTSORT:Boolean Algebra (Structure)}}

[[bg:Булева алгебра]]

[[Category:Boolean algebra| ]]

[[bn:বুলিয়ান বীজগণিত]]

[[Category:Algebraic structures]]

[[ca:Àlgebra de Boole]]

[[Category:Ockham algebras]]

[[cs:Booleova algebra]]

[[de:Boolesche Algebra]]

[[es:Álgebra de Boole]]

[[fa:جبر بولی]]

[[fr:Algèbre de Boole (logique)]]

[[gl:Álxebra de Boole]]

[[hr:Booleova algebra]]

[[io:Booleana algebro]]

[[id:Aljabar Boolean]]

[[it:Algebra di Boole]]

[[he:אלגברה בוליאנית]]

[[lt:Būlio algebra]]

[[nl:Booleaanse algebra]]

[[ja:ブール代数]]

[[pl:Algebra Boole'a]]

[[pt:Álgebra booleana]]

[[ru:Булева алгебра]]

[[sl:Booleova algebra]]

[[sr:Булова алгебра]]

[[sv:Boolesk algebra]]

[[th:พีชคณิตแบบบูล]]

[[tr:Boole cebiri]]

[[uk:Булева алгебра]]

[[zh:布尔代数]]