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In mathematics, a closure operator on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'' is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
\operatorname: \mathcal(S)\rightarrow \mathcal(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", in honor of E. H. Moore who studied closure operators in his 1910 ''Introduction to a form of general analysis'', whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor. Closure operators are also called "hull operators", which prevents confusion with the "closure operators" studied in topology. A set together with a closure operator on it is sometimes called a closure space.


Examples

Clearly, the usual set closure from topology is a closure operator. Other examples include the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteriz ...
of a subset of a vector space, the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...
or
affine hull In mathematics, the affine hull or affine span of a set ''S'' in Euclidean space R''n'' is the smallest affine set containing ''S'', or equivalently, the intersection of all affine sets containing ''S''. Here, an ''affine set'' may be defined a ...
of a subset of a vector space or the
lower semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...
hull \overline of a function f \colon E \to \mathbb \cup \, where E is e.g. a
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, defined implicitely \operatorname(\overline) = \overline, where \operatorname(f) is the epigraph of a function f. The relative interior \operatorname is not a closure operator: although it is idempotent, it is not increasing and if C_1 is a cube in \mathbb^3 and C_2 is one of its faces, then C_2 \subset C_1, but \operatorname(C_1) \ne \emptyset \ne \operatorname(C_2) and \operatorname(C_1) \cap \operatorname(C_2) = \emptyset, so it is not increasing.


Applications

Closure operators have many applications: In topology, the closure operators are ''topological'' closure operators, which must satisfy : \operatorname(X_1 \cup\dots\cup X_n) = \operatorname(X_1)\cup\dots\cup \operatorname(X_n) for all n\in\N (Note that for n=0 this gives \operatorname(\varnothing)=\varnothing). In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
and logic, many closure operators are finitary closure operators, i.e. they satisfy : \operatorname(X) = \bigcup\left\. In the theory of
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s, which are important in
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...
, closure operators have a more general definition that replaces \subseteq with \leq. (See .)


Closure operators in topology

The topological closure of a subset ''X'' of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
consists of all points ''y'' of the space, such that every neighbourhood of ''y'' contains a point of ''X''. The function that associates to every subset ''X'' its closure is a topological closure operator. Conversely, every topological closure operator on a set gives rise to a topological space whose closed sets are exactly the closed sets with respect to the closure operator.


Closure operators in algebra

Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called ''algebraic closure operators''. Every subset of an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
''generates'' a subalgebra: the smallest subalgebra containing the set. This gives rise to a finitary closure operator. Perhaps the best known example for this is the function that associates to every subset of a given vector space its
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteriz ...
. Similarly, the function that associates to every subset of a given
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
generated by it, and similarly for
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s and all other types of
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s. The linear span in a vector space and the similar algebraic closure in a field both satisfy the ''exchange property:'' If ''x'' is in the closure of the union of ''A'' and but not in the closure of ''A'', then ''y'' is in the closure of the union of ''A'' and . A finitary closure operator with this property is called a matroid. The dimension of a vector space, or the transcendence degree of a field (over its prime field) is exactly the rank of the corresponding matroid. The function that maps every subset of a given
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
to its
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
is also a finitary closure operator, and in general it is different from the operator mentioned before. Finitary closure operators that generalize these two operators are studied in model theory as dcl (for ''definable closure'') and acl (for ''algebraic closure''). The
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...
in ''n''-dimensional Euclidean space is another example of a finitary closure operator. It satisfies the ''anti-exchange property:'' If ''x'' is in the closure of the union of and ''A'', but not in the union of and closure of ''A'', then ''y'' is not in the closure of the union of and ''A''. Finitary closure operators with this property give rise to
antimatroid In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroid ...
s. As another example of a closure operator used in algebra, if some algebra has universe ''A'' and ''X'' is a set of pairs of ''A'', then the operator assigning to ''X'' the smallest congruence containing ''X'' is a finitary closure operator on ''A x A''.


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 cl(''X'') be the set of all formulas that can be derived from ''X''. Then cl is a closure operator on ''P''. More precisely, we can obtain cl as follows. Call "continuous" an operator ''J'' such that, for every
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''Di ...
class ''T'', :''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 Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and complete ...
(see Gerla 2000).


Consequence operators

Around 1930,
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of ''sentences''). In
abstract algebraic logic In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 200 ...
, finitary closure operators are still studied under the name ''consequence operator'', which was coined by Tarski. The set ''S'' represents a set of sentences, a subset ''T'' of ''S'' a theory, and cl(''T'') is the set of all sentences that follow from the theory. Nowadays the term can refer to closure operators that need not be finitary; finitary closure operators are then sometimes called finite consequence operators.


Closed and pseudo-closed sets

The closed sets with respect to a closure operator on ''S'' form a subset ''C'' of the power set ''P''(''S''). Any intersection of sets in ''C'' is again in ''C''. In other words, ''C'' is a complete meet-subsemilattice of ''P''(''S''). Conversely, if ''C'' ⊆ ''P''(''S'') is closed under arbitrary intersections, then the function that associates to every subset ''X'' of ''S'' the smallest set ''Y'' ∈ ''C'' such that ''X'' ⊆ ''Y'' is a closure operator. There is a simple and fast algorithm for generating all closed sets of a given closure operator. A closure operator on a set is topological if and only if the set of closed sets is closed under finite unions, i.e., ''C'' is a meet-complete sublattice of ''P''(''S''). Even for non-topological closure operators, ''C'' can be seen as having the structure of a lattice. (The join of two sets ''X'',''Y'' ⊆ ''P''(''S'') being cl(''X'' \cup ''Y'').) But then ''C'' is not a sublattice of the lattice ''P''(''S''). Given a finitary closure operator on a set, the closures of finite sets are exactly the
compact element {{Unreferenced, date=December 2008 In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not alr ...
s of the set ''C'' of closed sets. It follows that ''C'' is an
algebraic poset {{Unreferenced, date=December 2008 In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not alr ...
. Since ''C'' is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if ''C'' is an algebraic poset, then the closure operator is finitary. Each closure operator on a finite set ''S'' is uniquely determined by its images of its ''pseudo-closed'' sets. These are recursively defined: A set is pseudo-closed if it is not closed and contains the closure of each of its pseudo-closed proper subsets. Formally: ''P'' ⊆ ''S'' is pseudo-closed if and only if * ''P'' ≠ cl(''P'') and * if ''Q'' ⊂ ''P'' is pseudo-closed, then cl(''Q'') ⊆ ''P''.


Closure operators on partially ordered sets

A
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
(poset) is a set together with a ''partial order'' ≤, i.e. a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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 ''μA''(''X'') = ''A'' ∪ ''X'' is a closure operator, whereas ''λA''(''X'') = ''A'' ∩ ''X'' is a kernel operator. The
ceiling function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least i ...
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 In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If ''A'', ''B'' are posets, a function ''f'': ''A'' → ''B'' is defined to be monotone if it is ...
) 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.Blyth, p. 10 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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
, 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
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', an ...
s 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 In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
(join) operation might differ from that of ''P''). When ''P'' is the powerset
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
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 cl1 ≤ cl2
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
cl1(''x'') ≤ cl2(''x'') for all ''x'' in ''P''.


See also

* * * * * * *


Notes


References

* Garrett Birkhoff. 1967 (1940). ''Lattice Theory, 3rd ed''. American Mathematical Society. * Burris, Stanley N., and H.P. Sankappanavar (1981
A Course in Universal Algebra
Springer-Verlag. ''Free online edition''. * Brown, D.J. and Suszko, R. (1973) "Abstract Logics," Dissertationes Mathematicae 102- 9-42. * Castellini, G. (2003) ''Categorical closure operators''. Boston MA: Birkhaeuser. * Edelman, Paul H. (1980) ''Meet-distributive lattices and the anti-exchange closure,''
Algebra Universalis ''Algebra Universalis'' is an international scientific journal focused on universal algebra and lattice theory. The journal, founded in 1971 by George Grätzer, is currently published by Springer-Verlag. Honorary editors in chief of the journal ...
10: 290-299. * Ganter, Bernhard and Obiedkov, Sergei (2016) ''Conceptual Exploration''. Springer, . * Gerla, G. (2000) ''Fuzzy Logic: Mathematical Tools for Approximate Reasoning''.
Kluwer Academic Publishers Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...
. * Lloyd, J.W. (1987) ''Foundations of Logic Programming''. Springer-Verlag. * Tarski, Alfred (1983) "Fundamental concepts of the methodology of deductive sciences" in ''Logic, Semantics, Metamathematics''. Hackett (1956 ed., Oxford University Press). *
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
(1956) ''Logic, semantics and metamathematics''. Oxford University Press. * Ward, Morgan (1942) "The closure operators of a lattice,"
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
43: 191-96. * G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: ''Continuous Lattices and Domains'', Cambridge University Press, 2003 * * M. Erné, J. Koslowski, A. Melton, G. E. Strecker, ''A primer on Galois connections'', in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103–125. Available online in various file formats
PS.GZPS


External links

* Stanford Encyclopedia of Philosophy:
Algebraic Propositional Logic
—by Ramon Jansana. {{DEFAULTSORT:Closure Operator * Universal algebra Order theory pl:Operator konsekwencji