"Galois connection" meaning in All languages combined

{
  "etymology_text": "From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.",
  "forms": [
    {
      "form": "Galois connections",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Galois connection (plural Galois connections)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
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          "source": "w"
        },
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Category theory",
          "orig": "en:Category theory",
          "parents": [
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            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,\nDefine maps G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind."
        },
        {
          "ref": "2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:",
          "text": "Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of \"polarities,\" and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.",
          "type": "quote"
        },
        {
          "ref": "2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:",
          "text": "Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #x5C;left#x5C;langle#x5C;uparrow,#x5C;downarrow#x5C;right#x5C;rangle of mappings #x5C;uparrow assigning subcollections of Y to subcollections of X, and #x5C;downarrow assigning subcollections of X to subcollections of Y.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A type of correspondence between partially ordered sets (posets), also applicable to preordered sets."
      ],
      "hypernyms": [
        {
          "word": "adjunction"
        }
      ],
      "hyponyms": [
        {
          "word": "isotone Galois connection"
        },
        {
          "word": "monotone Galois connection"
        },
        {
          "word": "antitone Galois connection"
        }
      ],
      "id": "en-Galois_connection-en-noun-J9pfk4K9",
      "links": [
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          "category theory",
          "category theory"
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        [
          "correspondence",
          "correspondence"
        ],
        [
          "partially ordered set",
          "partially ordered set"
        ],
        [
          "poset",
          "poset"
        ],
        [
          "preordered set",
          "preordered set"
        ]
      ],
      "raw_glosses": [
        "(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets."
      ],
      "synonyms": [
        {
          "word": "Galois connexion"
        }
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "order-theory",
        "physical-sciences",
        "sciences"
      ],
      "wikipedia": [
        "Galois connection",
        "Transactions of the American Mathematical Society",
        "Évariste Galois",
        "Øystein Ore"
      ]
    }
  ],
  "word": "Galois connection"
}

{
  "etymology_text": "From Galois (attributive form of Galois theory) + connection; ultimately after French mathematician Évariste Galois. Coined by Norwegian mathematician Øystein Ore in 1944, Galois connexions, Transactions of the American Mathematical Society, 55, pages 493-513.",
  "forms": [
    {
      "form": "Galois connections",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Galois connection (plural Galois connections)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "adjunction"
    }
  ],
  "hyponyms": [
    {
      "word": "isotone Galois connection"
    },
    {
      "word": "monotone Galois connection"
    },
    {
      "word": "antitone Galois connection"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Category theory"
      ],
      "examples": [
        {
          "text": "1986, Horst Herrlich, Miroslav Hušek, Galois Connections, Austin Melton, Mathematical Foundation of Programming Semantics: International Conference, Proceedings, Springer, Lecture Notes in Computer Science: 239, page 122,\nDefine maps G:A→B and F:B→A by G(a)=y!∈!Y|∀x!∈!axρy and F(b)=x!∈!X|∀y!∈!bxρy. Then (F,G) is called a Galois connection of the first kind."
        },
        {
          "ref": "2001, J. Michael Dunn, Gary M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press, page 398:",
          "text": "Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of \"polarities,\" and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities.",
          "type": "quote"
        },
        {
          "ref": "2006, Radim Bělohlávek, Taťána Funioková, Vilém Vychodil, “Galois connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes”, in Bernd Reusch, editor, Computational Intelligence, Theory and Applications: International Conference, Proceedings, Springer,, page 205:",
          "text": "Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair #x5C;left#x5C;langle#x5C;uparrow,#x5C;downarrow#x5C;right#x5C;rangle of mappings #x5C;uparrow assigning subcollections of Y to subcollections of X, and #x5C;downarrow assigning subcollections of X to subcollections of Y.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A type of correspondence between partially ordered sets (posets), also applicable to preordered sets."
      ],
      "links": [
        [
          "category theory",
          "category theory"
        ],
        [
          "correspondence",
          "correspondence"
        ],
        [
          "partially ordered set",
          "partially ordered set"
        ],
        [
          "poset",
          "poset"
        ],
        [
          "preordered set",
          "preordered set"
        ]
      ],
      "raw_glosses": [
        "(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets."
      ],
      "topics": [
        "category-theory",
        "computing",
        "engineering",
        "mathematics",
        "natural-sciences",
        "order-theory",
        "physical-sciences",
        "sciences"
      ],
      "wikipedia": [
        "Galois connection",
        "Transactions of the American Mathematical Society",
        "Évariste Galois",
        "Øystein Ore"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "Galois connexion"
    }
  ],
  "word": "Galois connection"
}