"algebraic poset" meaning in English

{
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      "form": "algebraic posets",
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      "expansion": "algebraic poset (plural algebraic posets)",
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  "lang_code": "en",
  "pos": "noun",
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          "langcode": "en",
          "name": "Algebra",
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        {
          "text": "1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,\nA poset (P,<) is said to be algebraic if and only if\ni) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,\nii) for every x∈P, the set\nK_X:=y∈P|ycompact,y<x\nis non-empty and up-directed, and\nx= operatorname supK_x.\nA poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.\nConcerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when \"enough\" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.\nThe concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) \"algebraic lattice\" (cf. [9], [20], I-4)."
        },
        {
          "ref": "1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:",
          "text": "ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.\nA structure (D,#x5C;le#x5F;D,#x5C;cup#x5F;D) where (D,#x5C;le#x5F;D) is an algebraic poset and #x5C;cup#x5F;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#x5C;cup#x5F;Dd#x3D;d for all d#x5C;inD) will be called a nondeterministic algebraic poset.",
          "type": "quote"
        },
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          "ref": "1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:",
          "text": "It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #x7B;A#x2A;#x5C;cupA#x2A;#x7D;#x5C;sqrt#x7B;#x7D; in the prefix order; this poset enjoys the added condition that every element is compact.",
          "type": "quote"
        },
        {
          "ref": "1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:",
          "text": "The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A partially ordered set (poset) in which every element is the supremum of the compact elements below it."
      ],
      "id": "en-algebraic_poset-en-noun-XVqOh6C2",
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          "algebra",
          "algebra"
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          "partially ordered set",
          "partially ordered set"
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        [
          "supremum",
          "supremum"
        ],
        [
          "compact element",
          "compact element"
        ]
      ],
      "raw_glosses": [
        "(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it."
      ],
      "related": [
        {
          "word": "complete partial order"
        },
        {
          "word": "up-complete poset"
        },
        {
          "word": "up-directed"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "order-theory",
        "sciences"
      ]
    }
  ],
  "word": "algebraic poset"
}

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      "form": "algebraic posets",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "complete partial order"
    },
    {
      "word": "up-complete poset"
    },
    {
      "word": "up-directed"
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          "text": "1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,\nA poset (P,<) is said to be algebraic if and only if\ni) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,\nii) for every x∈P, the set\nK_X:=y∈P|ycompact,y<x\nis non-empty and up-directed, and\nx= operatorname supK_x.\nA poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.\nConcerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when \"enough\" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.\nThe concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) \"algebraic lattice\" (cf. [9], [20], I-4)."
        },
        {
          "ref": "1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:",
          "text": "ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.\nA structure (D,#x5C;le#x5F;D,#x5C;cup#x5F;D) where (D,#x5C;le#x5F;D) is an algebraic poset and #x5C;cup#x5F;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#x5C;cup#x5F;Dd#x3D;d for all d#x5C;inD) will be called a nondeterministic algebraic poset.",
          "type": "quote"
        },
        {
          "ref": "1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:",
          "text": "It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #x7B;A#x2A;#x5C;cupA#x2A;#x7D;#x5C;sqrt#x7B;#x7D; in the prefix order; this poset enjoys the added condition that every element is compact.",
          "type": "quote"
        },
        {
          "ref": "1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:",
          "text": "The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A partially ordered set (poset) in which every element is the supremum of the compact elements below it."
      ],
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        [
          "algebra",
          "algebra"
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          "partially ordered set",
          "partially ordered set"
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          "supremum",
          "supremum"
        ],
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          "compact element",
          "compact element"
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      ],
      "raw_glosses": [
        "(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "order-theory",
        "sciences"
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  "word": "algebraic poset"
}