See Boolean algebra on Wiktionary
{ "derived": [ { "_dis1": "42 21 37", "word": "free Boolean algebra" } ], "etymology_text": "Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.", "forms": [ { "form": "Boolean algebras", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "Boolean algebra (plural Boolean algebras)", "name": "en-noun" } ], "hypernyms": [ { "_dis1": "38 24 38", "sense": "Ockham algebra", "word": "distributive lattice" }, { "_dis1": "37 23 40", "sense": "Heyting algebra", "word": "residuated lattice" }, { "_dis1": "42 21 37", "word": "MV-algebra" } ], "hyponyms": [ { "_dis1": "42 21 37", "word": "complete Boolean algebra" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "_dis1": "42 21 37", "word": "Boolean lattice" }, { "_dis1": "42 21 37", "word": "Boolean ring" } ], "senses": [ { "categories": [ { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "34 25 41", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "38 28 34", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "37 26 37", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "34 29 37", "kind": "other", "name": "Terms with Czech translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 28 38", "kind": "other", "name": "Terms with French translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 27 38", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "30 34 36", "kind": "other", "name": "Terms with Hungarian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 31 34", "kind": "other", "name": "Terms with Japanese translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 34", "kind": "other", "name": "Terms with Korean translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Macedonian translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 31 35", "kind": "other", "name": "Terms with Mandarin translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Romanian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Russian translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Serbo-Croatian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 30 34", "kind": "other", "name": "Terms with Swedish translations", "parents": [], "source": "w+disamb" } ], "examples": [ { "text": "The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.", "type": "example" }, { "text": "A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.", "type": "example" } ], "glosses": [ "An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called \"complement\"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)" ], "id": "en-Boolean_algebra-en-noun-OMxlQble", "links": [ [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "idempotent", "idempotent" ], [ "involutory", "involutory" ], [ "monoid", "monoid" ], [ "distribute", "distribute" ] ], "raw_glosses": [ "(algebra) An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called \"complement\"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)" ], "topics": [ "algebra", "mathematics", "sciences" ], "translations": [ { "_dis1": "82 9 9", "code": "cmn", "lang": "Chinese Mandarin", "sense": "algebraic structure", "word": "布爾代數" }, { "_dis1": "82 9 9", "code": "cmn", "lang": "Chinese Mandarin", "roman": "Bù'ěr dàishù", "sense": "algebraic structure", "word": "布尔代数" }, { "_dis1": "82 9 9", "code": "cs", "lang": "Czech", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Booleova algebra" }, { "_dis1": "82 9 9", "code": "cs", "lang": "Czech", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "booleovská algebra" }, { "_dis1": "82 9 9", "code": "fr", "lang": "French", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algèbre de Boole" }, { "_dis1": "82 9 9", "code": "fr", "lang": "French", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algèbre booléenne" }, { "_dis1": "82 9 9", "code": "de", "lang": "German", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "boolesche Algebra" }, { "_dis1": "82 9 9", "code": "hu", "lang": "Hungarian", "sense": "algebraic structure", "word": "Boole-algebra" }, { "_dis1": "82 9 9", "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebra booleana" }, { "_dis1": "82 9 9", "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "masculine" ], "word": "reticolo booleano" }, { "_dis1": "82 9 9", "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebra di Boole" }, { "_dis1": "82 9 9", "code": "ja", "lang": "Japanese", "roman": "Būru-daisū", "sense": "algebraic structure", "word": "ブール代数" }, { "_dis1": "82 9 9", "code": "ko", "lang": "Korean", "roman": "Bul daesu", "sense": "algebraic structure", "word": "불 대수" }, { "_dis1": "82 9 9", "code": "ko", "lang": "Korean", "roman": "Buul daesu", "sense": "algebraic structure", "word": "부울 대수" }, { "_dis1": "82 9 9", "code": "mk", "lang": "Macedonian", "roman": "Bulova algébra", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Булова алге́бра" }, { "_dis1": "82 9 9", "code": "ro", "lang": "Romanian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebră booleană" }, { "_dis1": "82 9 9", "code": "ru", "lang": "Russian", "roman": "búleva álgebra", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "бу́лева а́лгебра" }, { "_dis1": "82 9 9", "code": "sh", "lang": "Serbo-Croatian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Booleova algebra" }, { "_dis1": "82 9 9", "code": "sv", "lang": "Swedish", "sense": "algebraic structure", "word": "Boolesk algebra" } ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Computing", "orig": "en:Computing", "parents": [ "Technology", "All topics", "Fundamental" ], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Logic", "orig": "en:Logic", "parents": [ "Formal sciences", "Philosophy", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "34 25 41", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "38 28 34", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "37 26 37", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "34 29 37", "kind": "other", "name": "Terms with Czech translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 28 38", "kind": "other", "name": "Terms with French translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 27 38", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "30 34 36", "kind": "other", "name": "Terms with Hungarian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 31 34", "kind": "other", "name": "Terms with Japanese translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 34", "kind": "other", "name": "Terms with Korean translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Macedonian translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 31 35", "kind": "other", "name": "Terms with Mandarin translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Romanian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Russian translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Serbo-Croatian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 30 34", "kind": "other", "name": "Terms with Swedish translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or \"true\" and \"false\") and are subject to operations based on AND, OR and NOT" ], "id": "en-Boolean_algebra-en-noun-ALQ6SjTq", "links": [ [ "algebra", "algebra" ], [ "logic", "logic" ], [ "computing", "computing#Noun" ], [ "element", "element" ], [ "values", "values" ], [ "operation", "operation" ], [ "AND", "AND" ], [ "OR", "OR" ], [ "NOT", "NOT" ] ], "raw_glosses": [ "(algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or \"true\" and \"false\") and are subject to operations based on AND, OR and NOT" ], "synonyms": [ { "_dis1": "1 99 0", "sense": "Specifically ...", "word": "switching algebra" } ], "topics": [ "algebra", "computing", "engineering", "human-sciences", "logic", "mathematics", "natural-sciences", "philosophy", "physical-sciences", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Mathematics", "orig": "en:Mathematics", "parents": [ "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "34 25 41", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "31 22 46", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "38 28 34", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "37 26 37", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "34 29 37", "kind": "other", "name": "Terms with Czech translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 28 38", "kind": "other", "name": "Terms with French translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 27 38", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "30 34 36", "kind": "other", "name": "Terms with Hungarian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" }, { "_dis": "35 31 34", "kind": "other", "name": "Terms with Japanese translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 34", "kind": "other", "name": "Terms with Korean translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Macedonian translations", "parents": [], "source": "w+disamb" }, { "_dis": "34 31 35", "kind": "other", "name": "Terms with Mandarin translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Romanian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 29 35", "kind": "other", "name": "Terms with Russian translations", "parents": [], "source": "w+disamb" }, { "_dis": "33 28 39", "kind": "other", "name": "Terms with Serbo-Croatian translations", "parents": [], "source": "w+disamb" }, { "_dis": "36 30 34", "kind": "other", "name": "Terms with Swedish translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "The study of such algebras; Boolean logic, classical logic." ], "id": "en-Boolean_algebra-en-noun-T~eTsbyR", "links": [ [ "mathematics", "mathematics" ], [ "Boolean logic", "Boolean logic" ], [ "classical logic", "classical logic" ] ], "raw_glosses": [ "(mathematics) The study of such algebras; Boolean logic, classical logic." ], "topics": [ "mathematics", "sciences" ] } ], "wikipedia": [ "Boolean algebra", "George Boole" ], "word": "Boolean algebra" }
{ "categories": [ "English countable nouns", "English entries with incorrect language header", "English eponyms", "English lemmas", "English multiword terms", "English nouns", "Entries with translation boxes", "Pages with 1 entry", "Pages with entries", "Terms with Czech translations", "Terms with French translations", "Terms with German translations", "Terms with Hungarian translations", "Terms with Italian translations", "Terms with Japanese translations", "Terms with Korean translations", "Terms with Macedonian translations", "Terms with Mandarin translations", "Terms with Romanian translations", "Terms with Russian translations", "Terms with Serbo-Croatian translations", "Terms with Swedish translations" ], "derived": [ { "word": "free Boolean algebra" } ], "etymology_text": "Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.", "forms": [ { "form": "Boolean algebras", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "Boolean algebra (plural Boolean algebras)", "name": "en-noun" } ], "hypernyms": [ { "sense": "Ockham algebra", "word": "distributive lattice" }, { "sense": "Heyting algebra", "word": "residuated lattice" }, { "word": "MV-algebra" } ], "hyponyms": [ { "word": "complete Boolean algebra" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "word": "Boolean lattice" }, { "word": "Boolean ring" } ], "senses": [ { "categories": [ "English terms with usage examples", "en:Algebra" ], "examples": [ { "text": "The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary operator: division into 30, and identity elements: 1 and 30, forms a Boolean algebra.", "type": "example" }, { "text": "A Boolean algebra is a De Morgan algebra which also satisfies the law of excluded middle and the law of noncontradiction.", "type": "example" } ], "glosses": [ "An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called \"complement\"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)" ], "links": [ [ "algebra", "algebra" ], [ "algebraic structure", "algebraic structure" ], [ "idempotent", "idempotent" ], [ "involutory", "involutory" ], [ "monoid", "monoid" ], [ "distribute", "distribute" ] ], "raw_glosses": [ "(algebra) An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called \"complement\"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ "en:Algebra", "en:Computing", "en:Logic" ], "glosses": [ "Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or \"true\" and \"false\") and are subject to operations based on AND, OR and NOT" ], "links": [ [ "algebra", "algebra" ], [ "logic", "logic" ], [ "computing", "computing#Noun" ], [ "element", "element" ], [ "values", "values" ], [ "operation", "operation" ], [ "AND", "AND" ], [ "OR", "OR" ], [ "NOT", "NOT" ] ], "raw_glosses": [ "(algebra, logic, computing) Specifically, an algebra in which all elements can take only one of two values (typically 0 and 1, or \"true\" and \"false\") and are subject to operations based on AND, OR and NOT" ], "topics": [ "algebra", "computing", "engineering", "human-sciences", "logic", "mathematics", "natural-sciences", "philosophy", "physical-sciences", "sciences" ] }, { "categories": [ "en:Mathematics" ], "glosses": [ "The study of such algebras; Boolean logic, classical logic." ], "links": [ [ "mathematics", "mathematics" ], [ "Boolean logic", "Boolean logic" ], [ "classical logic", "classical logic" ] ], "raw_glosses": [ "(mathematics) The study of such algebras; Boolean logic, classical logic." ], "topics": [ "mathematics", "sciences" ] } ], "synonyms": [ { "sense": "Specifically ...", "word": "switching algebra" } ], "translations": [ { "code": "cmn", "lang": "Chinese Mandarin", "sense": "algebraic structure", "word": "布爾代數" }, { "code": "cmn", "lang": "Chinese Mandarin", "roman": "Bù'ěr dàishù", "sense": "algebraic structure", "word": "布尔代数" }, { "code": "cs", "lang": "Czech", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Booleova algebra" }, { "code": "cs", "lang": "Czech", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "booleovská algebra" }, { "code": "fr", "lang": "French", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algèbre de Boole" }, { "code": "fr", "lang": "French", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algèbre booléenne" }, { "code": "de", "lang": "German", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "boolesche Algebra" }, { "code": "hu", "lang": "Hungarian", "sense": "algebraic structure", "word": "Boole-algebra" }, { "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebra booleana" }, { "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "masculine" ], "word": "reticolo booleano" }, { "code": "it", "lang": "Italian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebra di Boole" }, { "code": "ja", "lang": "Japanese", "roman": "Būru-daisū", "sense": "algebraic structure", "word": "ブール代数" }, { "code": "ko", "lang": "Korean", "roman": "Bul daesu", "sense": "algebraic structure", "word": "불 대수" }, { "code": "ko", "lang": "Korean", "roman": "Buul daesu", "sense": "algebraic structure", "word": "부울 대수" }, { "code": "mk", "lang": "Macedonian", "roman": "Bulova algébra", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Булова алге́бра" }, { "code": "ro", "lang": "Romanian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "algebră booleană" }, { "code": "ru", "lang": "Russian", "roman": "búleva álgebra", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "бу́лева а́лгебра" }, { "code": "sh", "lang": "Serbo-Croatian", "sense": "algebraic structure", "tags": [ "feminine" ], "word": "Booleova algebra" }, { "code": "sv", "lang": "Swedish", "sense": "algebraic structure", "word": "Boolesk algebra" } ], "wikipedia": [ "Boolean algebra", "George Boole" ], "word": "Boolean algebra" }
Download raw JSONL data for Boolean algebra meaning in All languages combined (6.7kB)
{ "called_from": "page/1412", "msg": "gloss may contain unhandled list items: An algebraic structure (Σ,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called \"complement\"), and 0 and 1 are nullary operators (i.e., constants), such that (Σ,∨,0) is a commutative monoid, (Σ,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)", "path": [ "Boolean algebra" ], "section": "English", "subsection": "noun", "title": "Boolean algebra", "trace": "" }
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-15 from the enwiktionary dump dated 2024-12-04 using wiktextract (8a39820 and 4401a4c). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.