"finitely generated" meaning in English

See finitely generated in All languages combined, or Wiktionary

Adjective

Etymology: From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically. Etymology templates: {{langname|en}} English, {{senseno|en|alex_top}} sense 1.5, {{langname|en}} English, {{senseno|en|alex_top}} sense 1.5 Head templates: {{en-adj|-}} finitely generated (not comparable)
  1. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    (most generally, category theory, of an object X in a locally small category 饾挒 that admits filtered colimits of monomorphism) Such that the functor operatorname Hom_( mathcal )C(X,路) preserves those filtered colimits of monomorphisms.
    Tags: not-comparable Categories (topical): Algebra, Category theory
    Sense id: en-finitely_generated-en-adj-qNs32htV Categories (other): English entries with incorrect language header Disambiguation of English entries with incorrect language header: 14 8 12 33 33 Topics: algebra, category-theory, computing, engineering, mathematics, natural-sciences, physical-sciences, sciences
  2. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    (less generally, but still covering most cases outside pure category theory, of an object X in a concrete category) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.
    Tags: not-comparable Categories (topical): Algebra
    Sense id: en-finitely_generated-en-adj-zspW7wCo Topics: algebra, mathematics, sciences
  3. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    (abstract algebra, of a group-like structure: a group, module, ring, monoid, etc.) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)
    Tags: abstract, not-comparable, usually Categories (topical): Algebra
    Sense id: en-finitely_generated-en-adj-g2ElOa49 Categories (other): English entries with incorrect language header Disambiguation of English entries with incorrect language header: 14 8 12 33 33 Topics: algebra, mathematics, sciences
  4. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    (ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R.
    Tags: not-comparable Categories (topical): Algebra
    Sense id: en-finitely_generated-en-adj-PbZ0zTUT Categories (other): English entries with incorrect language header Disambiguation of English entries with incorrect language header: 14 8 12 33 33 Topics: algebra, mathematics, sciences
  5. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
    (of a space) Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).
    Tags: not-comparable Categories (topical): Algebra
    Sense id: en-finitely_generated-en-adj-en:alex_top Categories (other): English entries with incorrect language header Disambiguation of English entries with incorrect language header: 14 8 12 33 33 Topics: algebra, mathematics, sciences

Alternative forms

Download JSON data for finitely generated meaning in English (9.3kB)

{
  "etymology_templates": [
    {
      "args": {
        "1": "en"
      },
      "expansion": "English",
      "name": "langname"
    },
    {
      "args": {
        "1": "en",
        "2": "alex_top"
      },
      "expansion": "sense 1.5",
      "name": "senseno"
    },
    {
      "args": {
        "1": "en"
      },
      "expansion": "English",
      "name": "langname"
    },
    {
      "args": {
        "1": "en",
        "2": "alex_top"
      },
      "expansion": "sense 1.5",
      "name": "senseno"
    }
  ],
  "etymology_text": "From the study of generators. The motivation for calling topologies satisfying sense 1.5 \"finitely generated\" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is \"generated\" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.",
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "finitely generated (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
      "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": "Category theory",
          "orig": "en:Category theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "14 8 12 33 33",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Such that the functor operatorname Hom_( mathcal )C(X,路) preserves those filtered colimits of monomorphisms."
      ],
      "id": "en-finitely_generated-en-adj-qNs32htV",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "category theory",
          "category theory"
        ],
        [
          "object",
          "object"
        ],
        [
          "locally small",
          "locally small"
        ],
        [
          "category",
          "category"
        ],
        [
          "admit",
          "admit"
        ],
        [
          "filtered colimit",
          "filtered colimit"
        ],
        [
          "monomorphism",
          "monomorphism"
        ],
        [
          "functor",
          "functor"
        ]
      ],
      "qualifier": "most generally",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(most generally, category theory, of an object X in a locally small category 饾挒 that admits filtered colimits of monomorphism) Such that the functor operatorname Hom_( mathcal )C(X,路) preserves those filtered colimits of monomorphisms."
      ],
      "raw_tags": [
        "of an object X in a locally small category 饾挒 that admits filtered colimits of monomorphism"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set."
      ],
      "id": "en-finitely_generated-en-adj-zspW7wCo",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "category theory",
          "category theory"
        ],
        [
          "concrete",
          "concrete"
        ],
        [
          "quotient",
          "quotient"
        ],
        [
          "set",
          "set"
        ],
        [
          "target",
          "target"
        ],
        [
          "regular",
          "regular"
        ],
        [
          "epimorphism",
          "epimorphism"
        ]
      ],
      "qualifier": "less generally; but still covering most cases outside pure category theory; less generally; but still covering most cases outside pure category theory",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(less generally, but still covering most cases outside pure category theory, of an object X in a concrete category) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set."
      ],
      "raw_tags": [
        "of an object X in a concrete category"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "14 8 12 33 33",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)"
      ],
      "id": "en-finitely_generated-en-adj-g2ElOa49",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "group",
          "group"
        ],
        [
          "module",
          "module"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "monoid",
          "monoid"
        ],
        [
          "generators",
          "generators"
        ],
        [
          "addition",
          "addition"
        ],
        [
          "scalar",
          "scalar"
        ]
      ],
      "qualifier": "module; ring; module; ring",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(abstract algebra, of a group-like structure: a group, module, ring, monoid, etc.) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)"
      ],
      "raw_tags": [
        "of a group-like structure"
      ],
      "tags": [
        "abstract",
        "not-comparable",
        "usually"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "14 8 12 33 33",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R.",
        "Finitely generated as a (left) module over R."
      ],
      "id": "en-finitely_generated-en-adj-PbZ0zTUT",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "ring theory",
          "ring theory"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "module",
          "module"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R."
      ],
      "raw_tags": [
        "of a (left) ideal in a ring R"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "14 8 12 33 33",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w+disamb"
        }
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open)."
      ],
      "id": "en-finitely_generated-en-adj-en:alex_top",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "space",
          "topological space"
        ],
        [
          "Equipped",
          "equip"
        ],
        [
          "intersection",
          "intersection"
        ],
        [
          "open set",
          "open set"
        ]
      ],
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(of a space) Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open)."
      ],
      "raw_tags": [
        "of a space"
      ],
      "senseid": [
        "en:alex_top"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "finitely generated"
}
{
  "categories": [
    "English adjectives",
    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
    "English uncomparable adjectives"
  ],
  "etymology_templates": [
    {
      "args": {
        "1": "en"
      },
      "expansion": "English",
      "name": "langname"
    },
    {
      "args": {
        "1": "en",
        "2": "alex_top"
      },
      "expansion": "sense 1.5",
      "name": "senseno"
    },
    {
      "args": {
        "1": "en"
      },
      "expansion": "English",
      "name": "langname"
    },
    {
      "args": {
        "1": "en",
        "2": "alex_top"
      },
      "expansion": "sense 1.5",
      "name": "senseno"
    }
  ],
  "etymology_text": "From the study of generators. The motivation for calling topologies satisfying sense 1.5 \"finitely generated\" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is \"generated\" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.",
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "finitely generated (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
      "categories": [
        "en:Algebra",
        "en:Category theory"
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Such that the functor operatorname Hom_( mathcal )C(X,路) preserves those filtered colimits of monomorphisms."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "category theory",
          "category theory"
        ],
        [
          "object",
          "object"
        ],
        [
          "locally small",
          "locally small"
        ],
        [
          "category",
          "category"
        ],
        [
          "admit",
          "admit"
        ],
        [
          "filtered colimit",
          "filtered colimit"
        ],
        [
          "monomorphism",
          "monomorphism"
        ],
        [
          "functor",
          "functor"
        ]
      ],
      "qualifier": "most generally",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(most generally, category theory, of an object X in a locally small category 饾挒 that admits filtered colimits of monomorphism) Such that the functor operatorname Hom_( mathcal )C(X,路) preserves those filtered colimits of monomorphisms."
      ],
      "raw_tags": [
        "of an object X in a locally small category 饾挒 that admits filtered colimits of monomorphism"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Algebra"
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "category theory",
          "category theory"
        ],
        [
          "concrete",
          "concrete"
        ],
        [
          "quotient",
          "quotient"
        ],
        [
          "set",
          "set"
        ],
        [
          "target",
          "target"
        ],
        [
          "regular",
          "regular"
        ],
        [
          "epimorphism",
          "epimorphism"
        ]
      ],
      "qualifier": "less generally; but still covering most cases outside pure category theory; less generally; but still covering most cases outside pure category theory",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(less generally, but still covering most cases outside pure category theory, of an object X in a concrete category) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set."
      ],
      "raw_tags": [
        "of an object X in a concrete category"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Algebra"
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)"
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "group",
          "group"
        ],
        [
          "module",
          "module"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "monoid",
          "monoid"
        ],
        [
          "generators",
          "generators"
        ],
        [
          "addition",
          "addition"
        ],
        [
          "scalar",
          "scalar"
        ]
      ],
      "qualifier": "module; ring; module; ring",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(abstract algebra, of a group-like structure: a group, module, ring, monoid, etc.) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)"
      ],
      "raw_tags": [
        "of a group-like structure"
      ],
      "tags": [
        "abstract",
        "not-comparable",
        "usually"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Algebra"
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R.",
        "Finitely generated as a (left) module over R."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "ring theory",
          "ring theory"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "module",
          "module"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(ring theory, of a (left) ideal in a ring R) Finitely generated as a (left) module over R."
      ],
      "raw_tags": [
        "of a (left) ideal in a ring R"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    },
    {
      "categories": [
        "en:Algebra"
      ],
      "glosses": [
        "In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "create",
          "create"
        ],
        [
          "describe",
          "describe"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "generator",
          "generator"
        ],
        [
          "space",
          "topological space"
        ],
        [
          "Equipped",
          "equip"
        ],
        [
          "intersection",
          "intersection"
        ],
        [
          "open set",
          "open set"
        ]
      ],
      "raw_glosses": [
        "(algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:",
        "(of a space) Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open)."
      ],
      "raw_tags": [
        "of a space"
      ],
      "senseid": [
        "en:alex_top"
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "finitely generated"
}

This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-16 from the enwiktionary dump dated 2024-05-02 using wiktextract (e268c0e and 304864d). 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.