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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.Givant and Paul Halmos, 2009, p. 20== History == 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 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 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 Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 ''Lattice Theory''. In the 1960s, Paul Cohen, Dana Scott, 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-tuple consisting of a set ''A'', equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (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 axioms hold:Davey, Priestley, 1990, p.109, 131, 144::Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.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'' = ''b'' ∧ ''a'' if and only if ''a'' ∨ ''b'' = ''b''.The relation ≤ defined by ''a'' ≤ ''b'' if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ''a'' ∧ ''b'' and the join ''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.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..
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.〔Givant and Paul Halmos, 2009, p. 20〕
== History ==
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 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 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 Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 ''Lattice Theory''. In the 1960s, Paul Cohen, Dana Scott, 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-tuple consisting of a set ''A'', equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (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 axioms hold:〔Davey, Priestley, 1990, p.109, 131, 144〕
::
Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.
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'' = ''b'' ∧ ''a'' if and only if ''a'' ∨ ''b'' = ''b''.
The relation ≤ defined by ''a'' ≤ ''b'' if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ''a'' ∧ ''b'' and the join ''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.
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.〔.〕
抄文引用元・出典: フリー百科事典『 Boolean algebra is a six-tuple consisting of a set ''A'', equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (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 axioms hold:Davey, Priestley, 1990, p.109, 131, 144::Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.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'' = ''b'' ∧ ''a'' if and only if ''a'' ∨ ''b'' = ''b''.The relation ≤ defined by ''a'' ≤ ''b'' if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ''a'' ∧ ''b'' and the join ''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.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..">ウィキペディア（Wikipedia）』
■Boolean algebra is a six-tuple consisting of a set ''A'', equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (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 axioms hold:Davey, Priestley, 1990, p.109, 131, 144::Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.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'' = ''b'' ∧ ''a'' if and only if ''a'' ∨ ''b'' = ''b''.The relation ≤ defined by ''a'' ≤ ''b'' if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ''a'' ∧ ''b'' and the join ''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.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..">ウィキペディアで「In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.Givant and Paul Halmos, 2009, p. 20== History == 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 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 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 Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 ''Lattice Theory''. In the 1960s, Paul Cohen, Dana Scott, 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-tuple consisting of a set ''A'', equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (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 axioms hold:Davey, Priestley, 1990, p.109, 131, 144::Note, however, that the absorption law can be excluded from the set of axioms as it can be derived by the other axioms.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'' = ''b'' ∧ ''a'' if and only if ''a'' ∨ ''b'' = ''b''.The relation ≤ defined by ''a'' ≤ ''b'' if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ''a'' ∧ ''b'' and the join ''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.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..」の詳細全文を読む
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