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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.)
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.
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 algebraUniversal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Basic idea From the point of view of universal algebra, an algebra is a set A together with a collection of operations on A. An n- ary operation on A.
The ceiling function from the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays to the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays, which assigns to every real x the smallest integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st not smaller than x, is a closure operator as well.
Given a closure operator C, a closed element of P is an element x that is a fixed pointSee also fixed-point arithmetic. In mathematics, a fixed point of a function is a point that is mapped to itself by the function. For example, if f is defined on the real numbers by f ''x x''2 − 3''x + 4, then 2 is a fixed point of f because f 2) 2. 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.
Every Galois connectionIn 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. 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.
Any partially ordered set P can be viewed as a categoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan, 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 monadIn category theory, a monad or triple is a type of functor, together with two associated natural transformations. Monads are important in the theory of pairs of adjoint functors. They are formally similar to monoids (hence the name) and generalize closures on the category P.
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 C1 ≤ C2 iff C1(x) ≤ C2(x) for all x in P.
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