In mathematics, given a partially ordered set (*P*, ≤), a **closure operator** on *P* is a function *C* : *P* → *P* with the following properties: For other meanings of mathematics or math, see mathematics (disambiguation). ...
In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
Partial plot of a function f. ...
In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ...
In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...
[edit] ## Examples
The name comes from the fact that forming the closure of subsets of a topological space has these properties if the set of all subsets is ordered by inclusion ⊆. (Note that the topological closure operator is *not* characterized by these properties however; see the Kuratowski closure axioms for a complete characterization.) In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ...
Another typical closure operator is the following: take a group *G* and for any subset *X* of *G*, let *C*(*X*) be the subgroup generated by *X*, i.e. the smallest subgroup of *G* containing *X*. Then *C* is a closure operator on the set of subsets of *G*, ordered by inclusion ⊆. Analogous examples can be given for the subspace generated by a given subset of a vector space, for the subfield generated by a given subset of a field, or indeed for the subalgebra generated by a given subset of any algebra in the sense of universal algebra. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
Screenshot (from SSCX Star Warzone). ...
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...
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 a closure operator as well. In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ...
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...
[edit] ## Closed elements; properties Given a closure operator *C*, a *closed element* of *P* is an element *x* that is a fixed point of *C*, or equivalently, that is in the image of *C*. If *a* is closed and *x* is arbitrary, then we have *x* ≤ *a* if and only if *C*(*x*) ≤ *a*. So *C*(*x*) is the smallest closed element that's greater than or equal to *x*. We see that *C* is uniquely determined by the set of closed elements. In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...
Every Galois connection 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 *C* can be described as follows: if *A* is the set of closed elements with respect to *C*, then *C* : *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. In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ...
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 Posets that has the additional **idempotent** and **extensive** properties. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ...
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*). The closure operators on *P* form themselves a complete lattice; the order on closure operators is defined by *C*_{1} ≤ *C*_{2} iff *C*_{1}(*x*) ≤ *C*_{2}(*x*) for all *x* in *P*. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
[edit] ## Closure operators in logic Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the set *F* of all possible formulas, and let *P* be the power set of *F*, ordered by ⊆. For a set *X* of formulas, let *C*(*X*) be the set of all formulas that can be derived from *X*. Then *C* is a closure operator on *P*. More precisely, if we call “continuous” an operator *J* such that, for every directed class *T*, *J(limT)=limJ(T),* then the deduction operator of a monotonic logic is a closure operator which is definable by a fixed point theorem from a continuous operator *J* (one-step consequence operator). In accordance, 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 fuzzy logic (see Gerla 2000). Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
[edit] ## Bibliography - Brown D.J. and Suszko R. (1973). Abstract Logics, Dissertationes Mathematicae, 102, 9-42.
- Castellini G. (2003) Categorial closure operators, Birkauser.
- Gerla G. (2000) Fuzzy Logic: Mathematical Tools for Approximate Reasoning, Kluwer Ac. Publishers.
- Lloyd J.W. (1987) Foundations of Logic Programming, Springer-Verlag, Berlin.
- Tarski A. (1956). Logic, semantics and metamathematics, Clarendon Press, Oxford.
- Ward M. (1942). The closure operators of a lattice, Annals of Mathematics, 43, 191-196.
[edit] ## See also |