About: Closure operator

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In mathematics, a closure operator on a set S is a function from the power set of S to itself that satisfies the following conditions for all sets Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families", in honor of E. H. Moore who studied closure operators in his 1910 Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekindand Ge

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rdf:type	yago:WikicatClosureOperators yago:Abstraction100002137 yago:Communication100033020 yago:Function113783816 yago:Language106282651 yago:MathematicalRelation113783581 yago:Operator113786413 yago:Relation100031921 yago:WikicatFormalLanguages
rdfs:label	Hüllenoperator (de) Closure operator (en) 폐포 연산자 (ko) Оператор замыкания (ru) 闭包算子 (zh)
rdfs:comment	In der Mathematik versteht man unter der Hülle einer Menge eine Obermenge, die groß genug ist, um bestimmte Anforderungen zu erfüllen, und zugleich die kleinste Menge ist, die diese Anforderungen erfüllt. Beispiele sind die konvexe Hülle einer Teilmenge eines Vektorraums, die abgeschlossene Hülle einer Teilmenge eines topologischen Raums oder die transitive Hülle einer zweistelligen Relation. Hüllenoperator bezeichnet die Vorschrift, durch die jeder Menge von Objekten ihre Hülle zugeordnet wird. Die durch einen Hüllenoperator gegebenen Hüllen bilden ein Hüllensystem, also ein Mengensystem mit bestimmten Eigenschaften. (de) 순서론에서 폐포 연산자(閉包演算子, 영어: closure operator) 또는 폐포 연산(閉包演算, 영어: closure operation)은 위상수학의 폐포와 유사한 성질들을 만족시키는 함수이다. 위상수학적 폐포와 달리 유한 합집합을 보존할 필요가 없다. 완비 격자를 판단하는 데 쓰일 수 있다. 보편 대수학과 계산 복잡도 이론 등에서 응용된다. (ko) 在数学中，给定偏序集合 (P, ≤)，在 P 上的闭包算子是函数 C : P → P 带有如下性质: * x ≤ C(x) 对于所有 x，就是说 C 是扩展性的。 * 如果 x ≤ y，则 C(x) ≤ C(y)，就是 C 是单调递增的。 * C(C(x)) = C(x) 对于所有的 x，就是说 C 是幂等函数。 (zh) In mathematics, a closure operator on a set S is a function from the power set of S to itself that satisfies the following conditions for all sets Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families", in honor of E. H. Moore who studied closure operators in his 1910 Introduction to a form of general analysis, whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekindand Ge (en) Оператор замыкания — обобщение интуитивной концепции замыкания. Именно: если — частично упорядоченное множество, оператор будет называться оператором замыкания, если выполнены три условия: , (монотонность) * (идемпотентность) В роли множества часто выступает булеан некоторого другого множества ; примеры этого можно найти в топологии, алгебре и логике. . Примеры операторов замыкания можно найти в самых разных областях математики: В топологии изучается замыкание множества. Топологическое замыкание «уважает» конечное объединение множеств: для любого . (ru)
dcterms:subject	Universal algebra Order theory Closure operators
Wikipage page ID	483120 (xsd:integer)
Wikipage revision ID	1122713473 (xsd:integer)
Link from a Wikipage to another Wikipage	Power set Prime field Model theory Monad (category theory) Morgan Ward Binary relation Algebra Universalis Algebraic closure Algebraic structure Universal algebra Annals of Mathematics Antimatroid Richard Dedekind Vector space Universal algebra Order theory Mathematical logic Mathematics Neighbourhood (mathematics) Residuated mapping Epigraph (mathematics) Frigyes Riesz Function (mathematics) Galois connection Garrett Birkhoff Georg Cantor Congruence relation Convex hull Logic Lower semicontinuous Stanford Encyclopedia of Philosophy Compact element Complete lattice Structure (mathematical logic) Subgroup Substructure (mathematics) Theoretical computer science Matroid Ceiling function Topology Duality (order theory) Fuzzy logic Linear span Alfred Tarski Algebra E. H. Moore Ernst Schröder (mathematician) Euclidean space Field (mathematics) Normed space Oxford University Press Partially ordered set Dimension (vector space) Directed set Group (mathematics) Fixpoint Abstract algebraic logic Affine hull Closure operators Supremum Relative interior Boolean algebra Idempotent Identity function Iff Integer Kluwer Academic Publishers Kuratowski closure axioms Category theory Real number Set (mathematics) Topological closure Infimum Pointwise order Topological space Transcendence degree

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