Closure operator

In mathematics, a closure operator on a set S is a function \operatorname{cl}: \mathcal{P}(S)\rightarrow \mathcal{P}(S) from the power set of S to itself that satisfies the following conditions for all sets X,Y\subseteq S : 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". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology.

In mathematics, a closure operator on a set S is a function \operatorname{cl}: \mathcal{P}(S)\rightarrow \mathcal{P}(S) from the power set of S to itself that satisfies the following conditions for all sets X,Y\subseteq S 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". A set together with a closure operator on it is sometimes called a closure space. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology. The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull \overline{f} of a function f \colon E \to \mathbb{R} \cup \{ \pm \infty \} , where E is e.g. a normed space, defined implicitly \operatorname{epi}(\overline{f}) = \overline{\operatorname{epi}(f)} , where \operatorname{epi}(f) is the epigraph of a function f . The relative interior \operatorname{ri} is not a closure operator: although it is idempotent, it is not increasing and if C_1 is a cube in \mathbb{R}^3 and C_2 is one of its faces, then C_2 \subset C_1 , but \operatorname{ri}(C_1) \ne \emptyset \ne \operatorname{ri}(C_2) and \operatorname{ri}(C_1) \cap \operatorname{ri}(C_2) = \emptyset , so it is not increasing. In topology, the closure operators are topological closure operators, which must satisfy \operatorname{cl}(X_1 \cup\dots\cup X_n) = \operatorname{cl}(X_1)\cup\dots\cup \operatorname{cl}(X_n) for all n\in\N (Note that for n=0 this gives \operatorname{cl}(\varnothing)=\varnothing ). In algebra and logic, many closure operators are finitary closure operators, i.e. they satisfy \operatorname{cl}(X) = \bigcup\left\{\operatorname{cl}(Y) : Y\subseteq X \text{ and } Y \text{ finite} \right\}. In the theory of partially ordered sets, which are important in theoretical computer science, closure operators have a more general definition that replaces \subseteq with \leq . (See .) J(lim T) = lim J(T). This continuity condition is on the basis of a fixed point theorem for J. Consider the one-step operator J of a monotone logic. This is the operator associating any set X of formulas with the set J(X) of formulas that are either logical axioms or are obtained by an inference rule from formulas in X or are in X. Then such an operator is continuous and we can define cl(X) as the least fixed point for J greater or equal to X. In accordance with such a point of view, Tarski, Brown, Suszko and other authors proposed a general approach to logic based on closure operator theory. Also, such an idea is proposed in programming logic (see Lloyd 1987) and in fuzzy logic (see Gerla 2000). A partially ordered set (poset) is a set together with a partial order ≤, i.e. a binary relation that is reflexive ( ), transitive ( implies ) and antisymmetric ( implies a = b). Every power set P(S) together with inclusion ⊆ is a partially ordered set. A function cl: P → P from a partial order P to itself is called a closure operator if it satisfies the following axioms for all elements x, y in P. More succinct alternatives are available: the definition above is equivalent to the single axiom x ≤ cl(y) if and only if cl(x) ≤ cl(y) for all x, y in P. Using the pointwise order on functions between posets, one may alternatively write the extensiveness property as id P ≤ cl, where id is the identity function. A self-map k that is increasing and idempotent, but satisfies the dual of the extensiveness property, i.e. k ≤ id P is called a kernel operator, interior operator, or dual closure. As examples, if A is a subset of a set B, then the self-map on the powerset of B given by μ A (X) = A ∪ X is a closure operator, whereas λ A (X) = A ∩ X is a kernel operator. The ceiling function from the real numbers to the real numbers, which assigns to every real x the smallest integer not smaller than x, is another example of a closure operator. A fixpoint of the function cl, i.e. an element c of P that satisfies cl(c) = c, is called a closed element. A closure operator on a partially ordered set is determined by its closed elements. If c is a closed element, then x ≤ c and cl(x) ≤ c are equivalent conditions. Every Galois connection (or residuated mapping) gives rise to a closure operator (as is explained in that article). In fact, every closure operator arises in this way from a suitable Galois connection. The Galois connection is not uniquely determined by the closure operator. One Galois connection that gives rise to the closure operator cl can be described as follows: if A is the set of closed elements with respect to cl, then cl: P → A is the lower adjoint of a Galois connection between P and A, with the upper adjoint being the embedding of A into P. Furthermore, every lower adjoint of an embedding of some subset into P is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint. Any partially ordered set P can be viewed as a category, with a single morphism from x to y if and only if x ≤ y. The closure operators on the partially ordered set P are then nothing but the monads on the category P. Equivalently, a closure operator can be viewed as an endofunctor on the category of partially ordered sets that has the additional idempotent and extensive properties. If P is a complete lattice, then a subset A of P is the set of closed elements for some closure operator on P if and only if A is a Moore family on P, i.e. the largest element of P is in A, and the infimum (meet) of any non-empty subset of A is again in A. Any such set A is itself a complete lattice with the order inherited from P (but the supremum (join) operation might differ from that of P). When P is the powerset Boolean algebra of a set X, then a Moore family on P is called a closure system on X. The closure operators on P form themselves a complete lattice; the order on closure operators is defined by cl 1 ≤ cl 2 iff cl 1 (x) ≤ cl 2 (x) for all x in P. ==See also== ==External links== Stanford Encyclopedia of Philosophy: "Algebraic Propositional Logic"—by Ramon Jansana. pl:Operator konsekwencji

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