See algebraic structure on Wiktionary
{
"forms": [
{
"form": "algebraic structures",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic structure (plural algebraic structures)",
"name": "en-noun"
}
],
"hyponyms": [
{
"_dis1": "0 0",
"sense": "ring",
"word": "division ring"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"glosses": [
"Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra."
],
"id": "en-algebraic_structure-en-noun-4upQaFvT",
"links": [
[
"algebra",
"algebra"
],
[
"universal algebra",
"universal algebra"
]
],
"raw_glosses": [
"(algebra, universal algebra) Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra."
],
"tags": [
"universal"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
{
"_dis": "19 81",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
},
{
"_dis": "10 90",
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "15 85",
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "23 77",
"kind": "other",
"name": "Terms with Czech translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "18 82",
"kind": "other",
"name": "Terms with Danish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "20 80",
"kind": "other",
"name": "Terms with French translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with German translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "23 77",
"kind": "other",
"name": "Terms with Hungarian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "18 82",
"kind": "other",
"name": "Terms with Icelandic translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with Macedonian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "22 78",
"kind": "other",
"name": "Terms with Norwegian Bokmål translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "17 83",
"kind": "other",
"name": "Terms with Norwegian Nynorsk translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with Portuguese translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with Romanian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "21 79",
"kind": "other",
"name": "Terms with Swedish translations",
"parents": [],
"source": "w+disamb"
}
],
"examples": [
{
"text": "The definition of algebraic structure establishes a framework through which to compare otherwise very different abstract objects, and to study similarities and patterns made apparent by the definition.",
"type": "example"
},
{
"text": "1975, Ronald James Williams, Algebraic Structures Up to Homotopy, University of California, San Diego, page 16,\nIn this chapter we propose a general means for describing algebraic structures, both strict and up to homotopy, and we apply the results of the preceding chapter to the study of these […] ."
},
{
"text": "1995, George R. Kempf, Algebraic Structures, Bertelsmann (Vieweg), page 129,\nThus the algebraic structures which we have been studying are part of category theory, but the important theorem is not generally categorical nonsense, although there are theorems in category theory which we will not study."
},
{
"ref": "2019, Palash B. Pal, A Physicist's Introduction to Algebraic Structures, Cambridge University Press, page 37:",
"text": "Any algebraic structure is defined by a non-empty set of elements, and a set of rules or conditions that the elements satisfy. Thus, even a set equipped with a relation, as described in Section 2.4, is an algebraic structure. This structure is sometimes called a setoid, a name that we won't have much use for.",
"type": "quote"
},
{
"ref": "2023, Michael Pilquist, Rúnar Bjarnason, Paul Chiusano, Functional programming with Scala, Second edition, Shelter Island: Manning Publications Co, →ISBN:",
"text": "A monoid is a purely algebraic structure consisting of an associative binary operation and an identity element for that operation.",
"type": "quote"
}
],
"glosses": [
"A mathematical object comprising a carrier set (aka underlying set or domain), an optional scalar set, a set of operations (typically binary operations, but otherwise each of finite arity) and a set of identities (axioms) which the operations must satisfy."
],
"hyponyms": [
{
"_dis1": "43 57",
"word": "setoid"
},
{
"_dis1": "43 57",
"word": "group"
},
{
"_dis1": "43 57",
"word": "vector space"
},
{
"_dis1": "43 57",
"word": "module"
},
{
"_dis1": "43 57",
"word": "integral domain"
},
{
"_dis1": "43 57",
"word": "field"
},
{
"_dis1": "43 57",
"word": "linear algebra"
},
{
"_dis1": "43 57",
"word": "associative algebra"
},
{
"_dis1": "43 57",
"word": "lattice"
},
{
"_dis1": "43 57",
"word": "category"
}
],
"id": "en-algebraic_structure-en-noun-wrmw1caD",
"links": [
[
"carrier set",
"carrier set"
],
[
"underlying set",
"underlying set"
],
[
"domain",
"domain"
],
[
"scalar set",
"scalar set"
],
[
"operation",
"operation"
],
[
"binary operation",
"binary operation"
],
[
"arity",
"arity"
],
[
"identities",
"identity"
],
[
"axiom",
"axiom"
]
],
"qualifier": "more formally",
"raw_glosses": [
"(more formally) A mathematical object comprising a carrier set (aka underlying set or domain), an optional scalar set, a set of operations (typically binary operations, but otherwise each of finite arity) and a set of identities (axioms) which the operations must satisfy."
],
"synonyms": [
{
"_dis1": "9 91",
"english": "in the context of universal algebra",
"sense": "formal structure of a carrier set, operations and axioms",
"word": "algebra"
},
{
"_dis1": "9 91",
"sense": "formal structure of a carrier set, operations and axioms",
"word": "algebraic system"
},
{
"_dis1": "9 91",
"sense": "formal structure of a carrier set, operations and axioms",
"word": "universal algebra"
}
],
"translations": [
{
"_dis1": "9 91",
"code": "cs",
"lang": "Czech",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "algebraická struktura"
},
{
"_dis1": "9 91",
"code": "da",
"lang": "Danish",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"common-gender"
],
"word": "algebraisk struktur"
},
{
"_dis1": "9 91",
"code": "fi",
"lang": "Finnish",
"sense": "formal structure of a carrier set, operations and identities",
"word": "algebrallinen rakenne"
},
{
"_dis1": "9 91",
"code": "fr",
"lang": "French",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "structure algébrique"
},
{
"_dis1": "9 91",
"code": "de",
"lang": "German",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "algebraische Struktur"
},
{
"_dis1": "9 91",
"code": "hu",
"lang": "Hungarian",
"sense": "formal structure of a carrier set, operations and identities",
"word": "algebrai struktúra"
},
{
"_dis1": "9 91",
"code": "is",
"lang": "Icelandic",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"neuter"
],
"word": "algebrumynstur"
},
{
"_dis1": "9 91",
"code": "is",
"lang": "Icelandic",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"neuter"
],
"word": "algebrulegt mynstur"
},
{
"_dis1": "9 91",
"code": "mk",
"lang": "Macedonian",
"roman": "algebarska struktura",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "алгебарска структура"
},
{
"_dis1": "9 91",
"code": "nb",
"lang": "Norwegian Bokmål",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"masculine"
],
"word": "algebraisk struktur"
},
{
"_dis1": "9 91",
"code": "nn",
"lang": "Norwegian Nynorsk",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"masculine"
],
"word": "algebraisk struktur"
},
{
"_dis1": "9 91",
"code": "pt",
"lang": "Portuguese",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "estrutura algébrica"
},
{
"_dis1": "9 91",
"code": "ro",
"lang": "Romanian",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "structură algebrică"
},
{
"_dis1": "9 91",
"code": "sv",
"lang": "Swedish",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"common-gender"
],
"word": "algebraisk struktur"
}
]
}
],
"wikipedia": [
"algebraic structure"
],
"word": "algebraic structure"
}
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Terms with Czech translations",
"Terms with Danish translations",
"Terms with Finnish translations",
"Terms with French translations",
"Terms with German translations",
"Terms with Hungarian translations",
"Terms with Icelandic translations",
"Terms with Macedonian translations",
"Terms with Norwegian Bokmål translations",
"Terms with Norwegian Nynorsk translations",
"Terms with Portuguese translations",
"Terms with Romanian translations",
"Terms with Swedish translations"
],
"forms": [
{
"form": "algebraic structures",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic structure (plural algebraic structures)",
"name": "en-noun"
}
],
"hyponyms": [
{
"word": "setoid"
},
{
"word": "group"
},
{
"word": "vector space"
},
{
"word": "module"
},
{
"sense": "ring",
"word": "division ring"
},
{
"word": "integral domain"
},
{
"word": "field"
},
{
"word": "linear algebra"
},
{
"word": "associative algebra"
},
{
"word": "lattice"
},
{
"word": "category"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"en:Algebra"
],
"glosses": [
"Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra."
],
"links": [
[
"algebra",
"algebra"
],
[
"universal algebra",
"universal algebra"
]
],
"raw_glosses": [
"(algebra, universal algebra) Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra."
],
"tags": [
"universal"
],
"topics": [
"algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"English terms with quotations",
"English terms with usage examples"
],
"examples": [
{
"text": "The definition of algebraic structure establishes a framework through which to compare otherwise very different abstract objects, and to study similarities and patterns made apparent by the definition.",
"type": "example"
},
{
"text": "1975, Ronald James Williams, Algebraic Structures Up to Homotopy, University of California, San Diego, page 16,\nIn this chapter we propose a general means for describing algebraic structures, both strict and up to homotopy, and we apply the results of the preceding chapter to the study of these […] ."
},
{
"text": "1995, George R. Kempf, Algebraic Structures, Bertelsmann (Vieweg), page 129,\nThus the algebraic structures which we have been studying are part of category theory, but the important theorem is not generally categorical nonsense, although there are theorems in category theory which we will not study."
},
{
"ref": "2019, Palash B. Pal, A Physicist's Introduction to Algebraic Structures, Cambridge University Press, page 37:",
"text": "Any algebraic structure is defined by a non-empty set of elements, and a set of rules or conditions that the elements satisfy. Thus, even a set equipped with a relation, as described in Section 2.4, is an algebraic structure. This structure is sometimes called a setoid, a name that we won't have much use for.",
"type": "quote"
},
{
"ref": "2023, Michael Pilquist, Rúnar Bjarnason, Paul Chiusano, Functional programming with Scala, Second edition, Shelter Island: Manning Publications Co, →ISBN:",
"text": "A monoid is a purely algebraic structure consisting of an associative binary operation and an identity element for that operation.",
"type": "quote"
}
],
"glosses": [
"A mathematical object comprising a carrier set (aka underlying set or domain), an optional scalar set, a set of operations (typically binary operations, but otherwise each of finite arity) and a set of identities (axioms) which the operations must satisfy."
],
"links": [
[
"carrier set",
"carrier set"
],
[
"underlying set",
"underlying set"
],
[
"domain",
"domain"
],
[
"scalar set",
"scalar set"
],
[
"operation",
"operation"
],
[
"binary operation",
"binary operation"
],
[
"arity",
"arity"
],
[
"identities",
"identity"
],
[
"axiom",
"axiom"
]
],
"qualifier": "more formally",
"raw_glosses": [
"(more formally) A mathematical object comprising a carrier set (aka underlying set or domain), an optional scalar set, a set of operations (typically binary operations, but otherwise each of finite arity) and a set of identities (axioms) which the operations must satisfy."
]
}
],
"synonyms": [
{
"english": "in the context of universal algebra",
"sense": "formal structure of a carrier set, operations and axioms",
"word": "algebra"
},
{
"sense": "formal structure of a carrier set, operations and axioms",
"word": "algebraic system"
},
{
"sense": "formal structure of a carrier set, operations and axioms",
"word": "universal algebra"
}
],
"translations": [
{
"code": "cs",
"lang": "Czech",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "algebraická struktura"
},
{
"code": "da",
"lang": "Danish",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"common-gender"
],
"word": "algebraisk struktur"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "formal structure of a carrier set, operations and identities",
"word": "algebrallinen rakenne"
},
{
"code": "fr",
"lang": "French",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "structure algébrique"
},
{
"code": "de",
"lang": "German",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "algebraische Struktur"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "formal structure of a carrier set, operations and identities",
"word": "algebrai struktúra"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"neuter"
],
"word": "algebrumynstur"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"neuter"
],
"word": "algebrulegt mynstur"
},
{
"code": "mk",
"lang": "Macedonian",
"roman": "algebarska struktura",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "алгебарска структура"
},
{
"code": "nb",
"lang": "Norwegian Bokmål",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"masculine"
],
"word": "algebraisk struktur"
},
{
"code": "nn",
"lang": "Norwegian Nynorsk",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"masculine"
],
"word": "algebraisk struktur"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "estrutura algébrica"
},
{
"code": "ro",
"lang": "Romanian",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"feminine"
],
"word": "structură algebrică"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "formal structure of a carrier set, operations and identities",
"tags": [
"common-gender"
],
"word": "algebraisk struktur"
}
],
"wikipedia": [
"algebraic structure"
],
"word": "algebraic structure"
}
Download raw JSONL data for algebraic structure meaning in All languages combined (6.9kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). 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.