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...

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...

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...

abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets...

axioms for its use in database theory Closure (mathematics), the result of applying a closure operator Closure (topology), for a set, the smallest closed...

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...

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...

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...

abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property...

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...

operator below or the article Kuratowski closure axioms. The interior operator int X {\displaystyle \operatorname {int} _{X}} is dual to the closure operator...

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...

transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operator is usually called transitive closure logic...

topological closure cl X ⁡ A {\displaystyle \operatorname {cl} _{X}A} satisfies the Kuratowski closure axioms. Conversely, for any closure operator A ↦ cl...

functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables...

and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator...

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...

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...

Google's Closure Templates, the Elvis operator is a null coalescing operator, equivalent to isNonnull($a) ? $a : $b. In Ballerina, the Elvis operator L ?:...

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...

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...

an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. Various...

deterministic transitive closure operators yield L, problems solvable in logarithmic space. First-order logic with a transitive closure operator yields NL, the...

in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets...

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...

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...

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...

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)...

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...

associated closure operator on subgroups of G {\displaystyle G} is H ¯ = H N {\displaystyle {\bar {H}}=HN} ; the associated kernel operator on subgroups...