"totally ordered" meaning in English

{
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "totally ordered (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with German translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "totally ordered set"
        }
      ],
      "examples": [
        {
          "text": "1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,\n(A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above)."
        },
        {
          "ref": "1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:",
          "text": "The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.",
          "type": "quote"
        },
        {
          "ref": "1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation",
          "roman": "ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.",
          "text": "G is a transitive group of order automorphisms of a totally ordered set X,\nθ is a convex G-congruence on the totally ordered set X,\nξ is the order type of the totally ordered set X,\n◌̅η is the order type of some class Y=xη of the congruence η,"
        }
      ],
      "glosses": [
        "That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset."
      ],
      "id": "en-totally_ordered-en-adj-puhfbGEa",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "total order",
          "total order"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "ground set",
          "ground set"
        ],
        [
          "partially ordered set",
          "partially ordered set"
        ],
        [
          "partial order",
          "partial order"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset."
      ],
      "related": [
        {
          "word": "chain"
        }
      ],
      "synonyms": [
        {
          "sense": "equipped with a total order",
          "word": "linearly ordered"
        }
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "that is equipped with a total order",
          "word": "total geordnet"
        }
      ],
      "wikipedia": [
        "Total order"
      ]
    }
  ],
  "word": "totally ordered"
}

{
  "derived": [
    {
      "word": "totally ordered set"
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "totally ordered (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "related": [
    {
      "word": "chain"
    }
  ],
  "senses": [
    {
      "categories": [
        "English adjectives",
        "English adverb-adjective phrases",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English terms with quotations",
        "English uncomparable adjectives",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with German translations",
        "en:Set theory"
      ],
      "examples": [
        {
          "text": "1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,\n(A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above)."
        },
        {
          "ref": "1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:",
          "text": "The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.",
          "type": "quote"
        },
        {
          "ref": "1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation",
          "roman": "ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.",
          "text": "G is a transitive group of order automorphisms of a totally ordered set X,\nθ is a convex G-congruence on the totally ordered set X,\nξ is the order type of the totally ordered set X,\n◌̅η is the order type of some class Y=xη of the congruence η,"
        }
      ],
      "glosses": [
        "That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "total order",
          "total order"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "ground set",
          "ground set"
        ],
        [
          "partially ordered set",
          "partially ordered set"
        ],
        [
          "partial order",
          "partial order"
        ]
      ],
      "raw_glosses": [
        "(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset."
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "mathematics",
        "order-theory",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "Total order"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "equipped with a total order",
      "word": "linearly ordered"
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "that is equipped with a total order",
      "word": "total geordnet"
    }
  ],
  "word": "totally ordered"
}