In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal...
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operations individually. The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations...
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interior operator. Let X {\displaystyle X} be an arbitrary set and ℘ ( X ) {\displaystyle \wp (X)} its power set. A Kuratowski closure operator is a unary...
30 KB (3,764 words) - 18:08, 9 October 2023
abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets...
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axioms for its use in database theory Closure (mathematics), the result of applying a closure operator Closure (topology), for a set, the smallest closed...
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Kleene star (redirect from Kleene closure)
mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of...
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Sequential space (redirect from Sequential closure operator)
that }}s_{\bullet }\to x\right\}} which defines a map, the sequential closure operator, on the power set of X . {\displaystyle X.} If necessary for clarity...
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Convex hull (redirect from Convex closure)
The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets...
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Idempotence (redirect from Idempotent operator)
abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property...
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Interior algebra (redirect from Closure algebra)
interior operator is the closure operator C defined by xC = ((x′)I)′. xC is called the closure of x. By the principle of duality, the closure operator satisfies...
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Interior (topology) (category Closure operators)
operator below or the article Kuratowski closure axioms. The interior operator int X {\displaystyle \operatorname {int} _{X}} is dual to the closure operator...
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a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these...
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transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operator is usually called transitive closure logic...
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topological closure cl X A {\displaystyle \operatorname {cl} _{X}A} satisfies the Kuratowski closure axioms. Conversely, for any closure operator A ↦ cl...
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functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables...
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Alexandrov topology (category Closure operators)
and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator...
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Sussman and Abelson also use the term closure in the 1980s with a second, unrelated meaning: the property of an operator that adds data to a data structure...
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topology, a preclosure operator or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not...
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Google's Closure Templates, the Elvis operator is a null coalescing operator, equivalent to isNonnull($a) ? $a : $b. In Ballerina, the Elvis operator L ?:...
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Fixed-point theorem (category Closure operators)
points. Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the...
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Galois connection (category Closure operators)
compositions GF : A → A, known as the associated closure operator, and FG : B → B, known as the associated kernel operator. Both are monotone and idempotent, and...
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an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. Various...
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deterministic transitive closure operators yield L, problems solvable in logarithmic space. First-order logic with a transitive closure operator yields NL, the...
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Matroid (category Closure operators)
in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets...
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topological spaces where we have no concrete way to measure distances. The closure operator closes a given set by mapping it to a closed set which contains the...
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In group theory, the normal closure of a subset S {\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle...
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General topology (section Closure operator definition)
also be determined by a closure operator (denoted cl), which assigns to any subset A ⊆ X its closure, or an interior operator (denoted int), which assigns...
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analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property)...
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finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm...
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associated closure operator on subgroups of G {\displaystyle G} is H ¯ = H N {\displaystyle {\bar {H}}=HN} ; the associated kernel operator on subgroups...
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