"well-order" meaning in English

{
  "forms": [
    {
      "form": "well-orders",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "well-order (plural well-orders)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "88 12",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
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          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "text": "1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,\n̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that\nf≨_(XY)g⇒h•f≨_(XY)h•g\nfor every h:Y→Z."
        },
        {
          "text": "2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,\nSome simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order."
        },
        {
          "text": "2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,\nDefinition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors).\nClearly the examples listed by von Neumann above, namely\nempty , empty, empty ,empty, empty ,empty,empty ,empty, …\nare all von Neumann well-orders if ordered by the membership relation \"∈,\" and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders."
        }
      ],
      "glosses": [
        "A total order of some set such that every nonempty subset contains a least element."
      ],
      "hypernyms": [
        {
          "sense": "partial order",
          "word": "preorder"
        }
      ],
      "id": "en-well-order-en-noun-0u6mLbbE",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "total order",
          "total order"
        ],
        [
          "nonempty",
          "nonempty"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "least",
          "least"
        ],
        [
          "element",
          "element"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory) A total order of some set such that every nonempty subset contains a least element."
      ],
      "synonyms": [
        {
          "sense": "type of total order",
          "word": "well-ordering"
        }
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "translations": [
        {
          "code": "da",
          "lang": "Danish",
          "sense": "a type of total order",
          "tags": [
            "common-gender"
          ],
          "word": "velordning"
        },
        {
          "code": "nl",
          "lang": "Dutch",
          "sense": "a type of total order",
          "tags": [
            "feminine"
          ],
          "word": "welordening"
        },
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "a type of total order",
          "word": "hyvinjärjestys"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "a type of total order",
          "tags": [
            "feminine"
          ],
          "word": "Wohlordnung"
        },
        {
          "code": "nb",
          "lang": "Norwegian Bokmål",
          "sense": "a type of total order",
          "tags": [
            "feminine",
            "masculine"
          ],
          "word": "velordning"
        },
        {
          "code": "nn",
          "lang": "Norwegian Nynorsk",
          "sense": "a type of total order",
          "tags": [
            "feminine"
          ],
          "word": "velordning"
        },
        {
          "code": "pl",
          "lang": "Polish",
          "sense": "a type of total order",
          "tags": [
            "masculine"
          ],
          "word": "porządek uporządkowany"
        },
        {
          "code": "sv",
          "lang": "Swedish",
          "sense": "a type of total order",
          "tags": [
            "common-gender"
          ],
          "word": "välordning"
        }
      ]
    }
  ],
  "synonyms": [
    {
      "_dis1": "0 0",
      "word": "well order"
    }
  ],
  "wikipedia": [
    "well-order"
  ],
  "word": "well-order"
}

{
  "forms": [
    {
      "form": "well-orders",
      "tags": [
        "present",
        "singular",
        "third-person"
      ]
    },
    {
      "form": "well-ordering",
      "tags": [
        "participle",
        "present"
      ]
    },
    {
      "form": "well-ordered",
      "tags": [
        "participle",
        "past"
      ]
    },
    {
      "form": "well-ordered",
      "tags": [
        "past"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)",
      "name": "en-verb"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "verb",
  "senses": [
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "The set of positive integers is well-ordered by the relation ≤.",
          "type": "example"
        },
        {
          "text": "1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111,\nStarting from these special well-ordered subsets, it is then possible to well-order the entire set."
        },
        {
          "text": "1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182,\nTo carry the analogy a bit further, the axiom of choice implies the ability to well order any set."
        },
        {
          "ref": "2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 18",
          "text": "Then #x5C;le#x5F;C is a well defined order on C, and (C,#x5C;le#x5F;C) belongs to #x5C;mathcal#x7B;X#x7D; (that is, #x5C;le#x5F;C well orders C) and is an upper bound for #x5C;mathcal#x7B;C#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "To impose a well-order on (a set)."
      ],
      "id": "en-well-order-en-verb-vKShPGkv",
      "links": [
        [
          "set theory",
          "set theory"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory, transitive) To impose a well-order on (a set)."
      ],
      "related": [
        {
          "word": "ordinal number"
        },
        {
          "word": "tree"
        }
      ],
      "tags": [
        "transitive"
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "Translations",
          "word": "hyvinjärjestää"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "Translations",
          "word": "wohlordnen"
        },
        {
          "code": "pl",
          "lang": "Polish",
          "sense": "Translations",
          "tags": [
            "imperfective"
          ],
          "word": "uporządkowywać"
        },
        {
          "code": "pl",
          "lang": "Polish",
          "sense": "Translations",
          "tags": [
            "perfective"
          ],
          "word": "uporządkować"
        }
      ]
    }
  ],
  "synonyms": [
    {
      "_dis1": "0 0",
      "word": "well order"
    }
  ],
  "wikipedia": [
    "well-order"
  ],
  "word": "well-order"
}

{
  "categories": [
    "English countable nouns",
    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
    "English nouns",
    "English verbs",
    "Translation table header lacks gloss"
  ],
  "forms": [
    {
      "form": "well-orders",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "well-order (plural well-orders)",
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  "hypernyms": [
    {
      "sense": "partial order",
      "word": "preorder"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "en:Set theory"
      ],
      "examples": [
        {
          "text": "1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,\n̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that\nf≨_(XY)g⇒h•f≨_(XY)h•g\nfor every h:Y→Z."
        },
        {
          "text": "2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,\nSome simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order."
        },
        {
          "text": "2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,\nDefinition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors).\nClearly the examples listed by von Neumann above, namely\nempty , empty, empty ,empty, empty ,empty,empty ,empty, …\nare all von Neumann well-orders if ordered by the membership relation \"∈,\" and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders."
        }
      ],
      "glosses": [
        "A total order of some set such that every nonempty subset contains a least element."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "total order",
          "total order"
        ],
        [
          "nonempty",
          "nonempty"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "least",
          "least"
        ],
        [
          "element",
          "element"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory) A total order of some set such that every nonempty subset contains a least element."
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "type of total order",
      "word": "well-ordering"
    },
    {
      "word": "well order"
    }
  ],
  "translations": [
    {
      "code": "da",
      "lang": "Danish",
      "sense": "a type of total order",
      "tags": [
        "common-gender"
      ],
      "word": "velordning"
    },
    {
      "code": "nl",
      "lang": "Dutch",
      "sense": "a type of total order",
      "tags": [
        "feminine"
      ],
      "word": "welordening"
    },
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "a type of total order",
      "word": "hyvinjärjestys"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "a type of total order",
      "tags": [
        "feminine"
      ],
      "word": "Wohlordnung"
    },
    {
      "code": "nb",
      "lang": "Norwegian Bokmål",
      "sense": "a type of total order",
      "tags": [
        "feminine",
        "masculine"
      ],
      "word": "velordning"
    },
    {
      "code": "nn",
      "lang": "Norwegian Nynorsk",
      "sense": "a type of total order",
      "tags": [
        "feminine"
      ],
      "word": "velordning"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "a type of total order",
      "tags": [
        "masculine"
      ],
      "word": "porządek uporządkowany"
    },
    {
      "code": "sv",
      "lang": "Swedish",
      "sense": "a type of total order",
      "tags": [
        "common-gender"
      ],
      "word": "välordning"
    }
  ],
  "wikipedia": [
    "well-order"
  ],
  "word": "well-order"
}

{
  "categories": [
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    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
    "English nouns",
    "English verbs",
    "Translation table header lacks gloss"
  ],
  "forms": [
    {
      "form": "well-orders",
      "tags": [
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        "singular",
        "third-person"
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    {
      "form": "well-ordering",
      "tags": [
        "participle",
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    {
      "form": "well-ordered",
      "tags": [
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    {
      "form": "well-ordered",
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  "head_templates": [
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      "args": {},
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "verb",
  "related": [
    {
      "word": "ordinal number"
    },
    {
      "word": "tree"
    }
  ],
  "senses": [
    {
      "categories": [
        "English terms with quotations",
        "English terms with usage examples",
        "English transitive verbs",
        "en:Set theory"
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      "examples": [
        {
          "text": "The set of positive integers is well-ordered by the relation ≤.",
          "type": "example"
        },
        {
          "text": "1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111,\nStarting from these special well-ordered subsets, it is then possible to well-order the entire set."
        },
        {
          "text": "1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182,\nTo carry the analogy a bit further, the axiom of choice implies the ability to well order any set."
        },
        {
          "ref": "2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 18",
          "text": "Then #x5C;le#x5F;C is a well defined order on C, and (C,#x5C;le#x5F;C) belongs to #x5C;mathcal#x7B;X#x7D; (that is, #x5C;le#x5F;C well orders C) and is an upper bound for #x5C;mathcal#x7B;C#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "To impose a well-order on (a set)."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory, transitive) To impose a well-order on (a set)."
      ],
      "tags": [
        "transitive"
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "well order"
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "Translations",
      "word": "hyvinjärjestää"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "Translations",
      "word": "wohlordnen"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "Translations",
      "tags": [
        "imperfective"
      ],
      "word": "uporządkowywać"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "Translations",
      "tags": [
        "perfective"
      ],
      "word": "uporządkować"
    }
  ],
  "wikipedia": [
    "well-order"
  ],
  "word": "well-order"
}