"Heyting algebra" meaning in English

{
  "etymology_text": "After Dutch mathematician Arend Heyting, who developed the theory as a way of modelling his intuitionistic logic.",
  "forms": [
    {
      "form": "Heyting algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Heyting algebra (plural Heyting algebras)",
      "name": "en-noun"
    }
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  "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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          "name": "English entries with language name categories using raw markup",
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          "parents": [
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
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            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "bi-Heyting algebra"
        },
        {
          "word": "biHeyting algebra"
        },
        {
          "word": "co-Heyting algebra"
        },
        {
          "word": "coHeyting algebra"
        },
        {
          "word": "complete Heyting algebra"
        }
      ],
      "examples": [
        {
          "ref": "1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii",
          "text": "The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.",
          "type": "quotation"
        },
        {
          "ref": "1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press, page 13, Proposition 1.2.14 should certainly be completed by the observation that the modus ponens holds as well in every Heyting algebra. Since, in the intuitionistic propositional calculus, being a true formula is being a terminal object (see proof of 1.1.3), the modus ponens of a Heyting algebra reduces to a=1 and a<b imply b=1",
          "text": "which is just obvious."
        },
        {
          "text": "1997, J. G. Stell, M. W. Worboys, The Algebraic Structure of Sets and Regions, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference, Proceedings, Springer, LNCS 1329, page 163,\nThe main contention of this paper is that Heyting algebras, and related structures, provide elegant and natural theories of parthood and boundary which have close connections to the above three ontologies."
        }
      ],
      "glosses": [
        "A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called \"implies\", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b)."
      ],
      "hypernyms": [
        {
          "sense": "bounded lattice",
          "word": "distributive lattice"
        },
        {
          "sense": "bounded lattice",
          "word": "residuated lattice"
        },
        {
          "sense": "bounded lattice",
          "word": "bicartesian closed category"
        }
      ],
      "hyponyms": [
        {
          "sense": "bounded lattice",
          "word": "Boolean algebra"
        },
        {
          "sense": "bounded lattice",
          "word": "complete Heyting algebra"
        },
        {
          "sense": "bounded lattice",
          "word": "finite distributive lattice"
        }
      ],
      "id": "en-Heyting_algebra-en-noun-DIWbcgel",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "bounded lattice",
          "bounded lattice"
        ],
        [
          "logical calculus",
          "logical calculus"
        ],
        [
          "binary operation",
          "binary operation"
        ],
        [
          "→",
          "→"
        ],
        [
          "⊃",
          "⊃"
        ],
        [
          "⇒",
          "⇒"
        ]
      ],
      "raw_glosses": [
        "(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called \"implies\", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b)."
      ],
      "related": [
        {
          "word": "Heyting prealgebra"
        },
        {
          "word": "pseudo-complement"
        },
        {
          "word": "relative pseudo-complement"
        },
        {
          "word": "residuated lattice"
        }
      ],
      "synonyms": [
        {
          "sense": "bounded lattice",
          "word": "pseudo-Boolean algebra"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "order-theory",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "bounded lattice equipped with operation called implies",
          "tags": [
            "feminine"
          ],
          "word": "algebra di Heyting"
        }
      ],
      "wikipedia": [
        "Arend Heyting",
        "Heyting algebra",
        "intuitionistic logic"
      ]
    }
  ],
  "word": "Heyting algebra"
}

{
  "derived": [
    {
      "word": "bi-Heyting algebra"
    },
    {
      "word": "biHeyting algebra"
    },
    {
      "word": "co-Heyting algebra"
    },
    {
      "word": "coHeyting algebra"
    },
    {
      "word": "complete Heyting algebra"
    }
  ],
  "etymology_text": "After Dutch mathematician Arend Heyting, who developed the theory as a way of modelling his intuitionistic logic.",
  "forms": [
    {
      "form": "Heyting algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Heyting algebra (plural Heyting algebras)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "sense": "bounded lattice",
      "word": "distributive lattice"
    },
    {
      "sense": "bounded lattice",
      "word": "residuated lattice"
    },
    {
      "sense": "bounded lattice",
      "word": "bicartesian closed category"
    }
  ],
  "hyponyms": [
    {
      "sense": "bounded lattice",
      "word": "Boolean algebra"
    },
    {
      "sense": "bounded lattice",
      "word": "complete Heyting algebra"
    },
    {
      "sense": "bounded lattice",
      "word": "finite distributive lattice"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Heyting prealgebra"
    },
    {
      "word": "pseudo-complement"
    },
    {
      "word": "relative pseudo-complement"
    },
    {
      "word": "residuated lattice"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "Quotation templates to be cleaned",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii",
          "text": "The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.",
          "type": "quotation"
        },
        {
          "ref": "1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press, page 13, Proposition 1.2.14 should certainly be completed by the observation that the modus ponens holds as well in every Heyting algebra. Since, in the intuitionistic propositional calculus, being a true formula is being a terminal object (see proof of 1.1.3), the modus ponens of a Heyting algebra reduces to a=1 and a<b imply b=1",
          "text": "which is just obvious."
        },
        {
          "text": "1997, J. G. Stell, M. W. Worboys, The Algebraic Structure of Sets and Regions, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference, Proceedings, Springer, LNCS 1329, page 163,\nThe main contention of this paper is that Heyting algebras, and related structures, provide elegant and natural theories of parthood and boundary which have close connections to the above three ontologies."
        }
      ],
      "glosses": [
        "A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called \"implies\", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
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        [
          "bounded lattice",
          "bounded lattice"
        ],
        [
          "logical calculus",
          "logical calculus"
        ],
        [
          "binary operation",
          "binary operation"
        ],
        [
          "→",
          "→"
        ],
        [
          "⊃",
          "⊃"
        ],
        [
          "⇒",
          "⇒"
        ]
      ],
      "raw_glosses": [
        "(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called \"implies\", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "order-theory",
        "sciences"
      ],
      "wikipedia": [
        "Arend Heyting",
        "Heyting algebra",
        "intuitionistic logic"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "bounded lattice",
      "word": "pseudo-Boolean algebra"
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "bounded lattice equipped with operation called implies",
      "tags": [
        "feminine"
      ],
      "word": "algebra di Heyting"
    }
  ],
  "word": "Heyting algebra"
}