About: Compact element

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In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.)

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  • In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.) (en)
  • 順序理論において、半順序集合上のある要素が、その要素以上の元を持たない非空有向集合の上限に含まれないとき、コンパクトあるいは有限であるという。 コンパクト性の概念は他にもあり、特に通常の集合論的な意味での「有限」という用語は順序理論においての「有限要素」の概念とは両立しないことに注意。 (ja)
  • 在数学领域的序理论中，偏序集合的紧致或有限元素是还未包含在紧致元素之上的成员的任何非空有向子集的上确界所不能包容的那些元素。 注意在数学中还有其他的紧致性概念，还有在常见的集合论中的术语有限的意义不一致于序理论的“有限元素”的概念。 (zh)
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  • In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.) (en)
  • 順序理論において、半順序集合上のある要素が、その要素以上の元を持たない非空有向集合の上限に含まれないとき、コンパクトあるいは有限であるという。 コンパクト性の概念は他にもあり、特に通常の集合論的な意味での「有限」という用語は順序理論においての「有限要素」の概念とは両立しないことに注意。 (ja)
  • 在数学领域的序理论中，偏序集合的紧致或有限元素是还未包含在紧致元素之上的成员的任何非空有向子集的上确界所不能包容的那些元素。 注意在数学中还有其他的紧致性概念，还有在常见的集合论中的术语有限的意义不一致于序理论的“有限元素”的概念。 (zh)
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  • Compact element (en)
  • コンパクト要素 (ja)
  • 紧致元素 (zh)
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