"antichain" meaning in English

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  "etymology_text": "From anti- + chain.",
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  "pos": "noun",
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      "derived": [
        {
          "word": "maximal antichain"
        }
      ],
      "examples": [
        {
          "ref": "1997, Winfried Just, Martin Weese, Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician, American Mathematical Society, page 140:",
          "text": "First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.",
          "type": "quote"
        },
        {
          "text": "2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, LNCS 7730, page 245,\nWe first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals."
        },
        {
          "text": "2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199,\nIn 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set N=1,2,3,…,n. Call a family ℱ of subsets of N, which has partial order ⊆]) an antichain if no set of ℱ contains another set of the family ℱ. What is the size of a largest antichain? Clearly, the family ℱₖ of all k-sets satisfies the antichain property with |ℱₖ|=\\binomnk. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size \\binomn[n/2]= max ₖ\\binomnk. Sperner's theorem now asserts that there are no larger ones."
        }
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        "A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤."
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      "id": "en-antichain-en-noun-9QBlfRm7",
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          "set theory",
          "set theory"
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          "graph theory",
          "graph theory"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "partially ordered set",
          "partially ordered set"
        ],
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          "element",
          "element"
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          "comparable",
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        "(set theory, order theory, graph theory) A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤."
      ],
      "topics": [
        "graph-theory",
        "mathematics",
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      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "subset of a partially ordered set",
          "word": "antiketju"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "subset of a partially ordered set",
          "tags": [
            "feminine"
          ],
          "word": "anticatena"
        },
        {
          "code": "es",
          "lang": "Spanish",
          "sense": "subset of a partially ordered set",
          "tags": [
            "feminine"
          ],
          "word": "anticadena"
        }
      ],
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        "antichain"
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  "word": "antichain"
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      "word": "maximal antichain"
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          "text": "First of all, Zorn's Lemma implies that every uncountable antichain in T is contained in a maximal uncountable antichain. So it suffices to make sure that T has no maximal uncountable antichains.",
          "type": "quote"
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          "text": "2013, Vijay K. Garg, Maximal Antichain Lattice Algorithms for Distributed Computations, Proceedings, Davide Frey, Michel Raynal, Saswati Sarkar, Rudrapatna K. Shyamasundar, Prasun Sinha (editors), Distributed Computing and Networking: 14th International Conference, ICDCN, Springer, LNCS 7730, page 245,\nWe first define three different but isomorphic lattices: the lattice of maximal antichain ideals, the lattice of maximal antichains and the lattice of strict ideals."
        },
        {
          "text": "2014, Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK, Springer, 5th Edition, page 199,\nIn 1928 Emanuel Sperner asked and answered the following question: Suppose we are given the set N=1,2,3,…,n. Call a family ℱ of subsets of N, which has partial order ⊆]) an antichain if no set of ℱ contains another set of the family ℱ. What is the size of a largest antichain? Clearly, the family ℱₖ of all k-sets satisfies the antichain property with |ℱₖ|=\\binomnk. Looking at the maximum of the binomial coefficients (see page 14) we conclude that there is an antichain of size \\binomn[n/2]= max ₖ\\binomnk. Sperner's theorem now asserts that there are no larger ones."
        }
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      ],
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        "(set theory, order theory, graph theory) A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤."
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  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "subset of a partially ordered set",
      "word": "antiketju"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "subset of a partially ordered set",
      "tags": [
        "feminine"
      ],
      "word": "anticatena"
    },
    {
      "code": "es",
      "lang": "Spanish",
      "sense": "subset of a partially ordered set",
      "tags": [
        "feminine"
      ],
      "word": "anticadena"
    }
  ],
  "word": "antichain"
}