See direct product in All languages combined, or Wiktionary
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corresponding original ring operations." ], "id": "en-direct_product-en-noun-mX5JRjwk", "links": [ [ "ring", "ring" ] ], "qualifier": "ring theory", "raw_glosses": [ "(ring theory) Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations." ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Topology", "orig": "en:Topology", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "13 21 13 24 14 15", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "12 13 10 24 22 18", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "16 19 13 22 16 14", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "15 19 14 21 17 13", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "17 14 12 20 22 15", "kind": "other", "name": "Terms with Finnish translations", "parents": [], "source": "w+disamb" }, { "_dis": "17 13 12 21 22 14", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "A topological space analogously formed from two or more (up to an infinite number of) topological spaces." ], "id": "en-direct_product-en-noun-mybMzLVx", "links": [ [ "topology", "topology" ], [ "topological space", "topological space" ] ], "raw_glosses": [ "(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces." ], "topics": [ "mathematics", "sciences", "topology" ], "translations": [ { "_dis1": "17 16 8 24 18 18", "code": "fi", "lang": "Finnish", "sense": "product of sets — see also Cartesian product", "word": "suora tulo" }, { "_dis1": "17 16 8 24 18 18", "code": "it", "lang": "Italian", "sense": "product of sets — see also Cartesian product", "tags": [ "masculine" ], "word": "prodotto diretto" } ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Mathematics", "orig": "en:Mathematics", "parents": [ "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "13 21 13 24 14 15", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "12 13 10 24 22 18", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "16 19 13 22 16 14", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "15 19 14 21 17 13", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "17 14 12 20 22 15", 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Itô, An Introduction to Probability Theory, page 53", "text": "Let us start with the definition of the direct product of two probability measures. Let P#95;1 and P#95;2 be probability measures on #92;Omega#95;1 and #92;Omega#95;2, respectively, and denote #92;Omega#95;1#92;times#92;Omega#95;2 by #92;Omega. A probability measure P on #92;Omega with #92;mathfrak#123;D#125;#92;left(P#92;right)#61;#92;mathfrak#123;D#125;#92;left(P#95;1#92;right)#92;times#92;mathfrak#123;D#125;#92;left(P#95;2#92;right) is called the direct product of P#95;1 and P#95;2 (written P#95;1#92;timesP#95;2) if\nP(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.\nThe probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written\n(Ω,P)=(Ω₁,P_1)×(Ω₂,P_2).\nFor example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.", "type": "quotation" } ], "glosses": [ "Any of a number of mathematical objects analogously derived from a given ordered set of objects." ], "id": "en-direct_product-en-noun-S-8OU84X", "links": [ [ "mathematics", "mathematics" ] ], "raw_glosses": [ "(mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects." ], "topics": [ "mathematics", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Category theory", "orig": "en:Category theory", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "13 21 13 24 14 15", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "12 13 10 24 22 18", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "16 19 13 22 16 14", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "15 19 14 21 17 13", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "17 14 12 20 22 15", "kind": "other", "name": "Terms with Finnish translations", "parents": [], "source": "w+disamb" }, { "_dis": "17 13 12 21 22 14", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" } ], "glosses": [ "A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects." ], "id": "en-direct_product-en-noun-3kiaMZq2", "links": [ [ "category theory", "category theory" ], [ "object", "object" ], [ "category", "category" ], [ "morphism", "morphism" ] ], "raw_glosses": [ "(category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects." ], "related": [ { "_dis1": "33 0 0 0 0 67", "english": "synonymous in specific domains", "word": "direct sum" }, { "_dis1": "33 0 0 0 0 67", "word": "product ring" }, { "_dis1": "33 0 0 0 0 67", "word": "product space" }, { "_dis1": "33 0 0 0 0 67", "word": "product topology" } ], "synonyms": [ { "_dis1": "13 8 3 6 22 49", "sense": "product of objects in a category", "word": "categorical product" }, { "_dis1": "13 8 3 6 22 49", "sense": "product of objects in a category", "word": "product" } ], "topics": [ "category-theory", "computing", "engineering", "mathematics", "natural-sciences", "physical-sciences", "sciences" ] } ], "wikipedia": [ "box topology", "direct product" ], "word": "direct product" }
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A periodic Abelian group is the direct product of its Sylow subgroups, S(p).", "type": "quote" } ], "glosses": [ "Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups." ], "links": [ [ "group theory", "group theory" ], [ "group", "group" ], [ "normal subgroup", "normal subgroup" ] ], "raw_glosses": [ "(group theory) Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups." ], "topics": [ "group-theory", "mathematics", "sciences" ] }, { "categories": [ "English terms with usage examples" ], "examples": [ { "text": "A Boolean ring of order 2ⁿ (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one.", "type": "example" } ], "glosses": [ "Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations." ], "links": [ [ "ring", "ring" ] ], "qualifier": "ring theory", "raw_glosses": [ "(ring theory) Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations." ] }, { "categories": [ "en:Topology" ], "glosses": [ "A topological space analogously formed from two or more (up to an infinite number of) topological spaces." ], "links": [ [ "topology", "topology" ], [ "topological space", "topological space" ] ], "raw_glosses": [ "(topology) A topological space analogously formed from two or more (up to an infinite number of) topological spaces." ], "topics": [ "mathematics", "sciences", "topology" ] }, { "categories": [ "English terms with quotations", "Quotation templates to be cleaned", "en:Mathematics" ], "examples": [ { "ref": "1978, K. Itô, An Introduction to Probability Theory, page 53", "text": "Let us start with the definition of the direct product of two probability measures. Let P#95;1 and P#95;2 be probability measures on #92;Omega#95;1 and #92;Omega#95;2, respectively, and denote #92;Omega#95;1#92;times#92;Omega#95;2 by #92;Omega. A probability measure P on #92;Omega with #92;mathfrak#123;D#125;#92;left(P#92;right)#61;#92;mathfrak#123;D#125;#92;left(P#95;1#92;right)#92;times#92;mathfrak#123;D#125;#92;left(P#95;2#92;right) is called the direct product of P#95;1 and P#95;2 (written P#95;1#92;timesP#95;2) if\nP(B_1×B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.\nThe probability space (Ω,P) is called the direct product of (Ω₁,P_1) and (Ω₂,P_2), written\n(Ω,P)=(Ω₁,P_1)×(Ω₂,P_2).\nFor example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.", "type": "quotation" } ], "glosses": [ "Any of a number of mathematical objects analogously derived from a given ordered set of objects." ], "links": [ [ "mathematics", "mathematics" ] ], "raw_glosses": [ "(mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects." ], "topics": [ "mathematics", "sciences" ] }, { "categories": [ "en:Category theory" ], "glosses": [ "A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects." ], "links": [ [ "category theory", "category theory" ], [ "object", "object" ], [ "category", "category" ], [ "morphism", "morphism" ] ], "raw_glosses": [ "(category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects." ], "topics": [ "category-theory", "computing", "engineering", "mathematics", "natural-sciences", "physical-sciences", "sciences" ] } ], "synonyms": [ { "sense": "product of sets", "word": "Cartesian product" }, { "sense": "product of objects in a category", "word": "categorical product" }, { "sense": "product of objects in a category", "word": "product" } ], "translations": [ { "code": "fi", "lang": "Finnish", "sense": "product of sets — see also Cartesian product", "word": "suora tulo" }, { "code": "it", "lang": "Italian", "sense": "product of sets — see also Cartesian product", "tags": [ "masculine" ], "word": "prodotto diretto" } ], "wikipedia": [ "box topology", "direct product" ], "word": "direct product" }
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