Closure operator: Difference between revisions
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In [[mathematics]] a '''closure operator''' is a [[unary operator]] or [[function (mathematics)|function]] on subsets of a given set which maps a subset to a containing subset with a particular property. | In [[mathematics]] a '''closure operator''' is a [[unary operator]] or [[function (mathematics)|function]] on subsets of a given set which maps a subset to a containing subset with a particular property. | ||
A ''closure operator'' on a set ''X'' is a function '' | A ''closure operator'' on a set ''X'' is a function ''F'' on the [[power set]] of ''X'', <math>F : \mathcal{P}X \rarr \mathcal{P}X</math>, satisfying: | ||
:<math>A \subseteq B \Rightarrow | :<math>A \subseteq B \Rightarrow FA \subseteq FB ;\,</math> | ||
:<math>A \subseteq | :<math>A \subseteq FA ;\,</math> | ||
:<math> | :<math>FFA = FA .\,</math> | ||
A ''topological closure operator'' satisfies the further property | A ''topological closure operator'' satisfies the further property | ||
:<math> | :<math>F(A \cup B) = FA \cup FB .\,</math> | ||
A ''closed'' set for ''F'' is one of the sets in the image of ''F'' | |||
==Closure system== | |||
A '''closure system''' is the set of closed sets of a closure operator. A closure system is defined as a family <math>\mathcal{C}</math> of subsets of a set ''X'' which contains ''X'' and is closed under taking arbitrary [[intersection]]s: | |||
:<math>\mathcal{S} \subseteq \mathcal{C} \Rightarrow \cap \mathcal{S} \in \mathcal{C} .\,</math> | |||
The closure operator ''F'' may be recovered from the closure system as | |||
:<math>FA = \bigcap_{A \subseteq C \in \mathcal{C}} C .\,</math> | |||
==Examples== | |||
In many [[algebraic structure]]s the set of substructures forms a closure system. The corresponding closure operator is often written <math>\langle A \rangle</math> and termed the substructure "generated" or "spanned" by ''A''. Notable examples include | |||
* Subsemigroups of a [[semigroup]] ''S''. The semigroup generated by a subset ''A'' may also be obtained as the set of all finite products of one or more elements of ''A''. | |||
* [[Subgroup]]s of a [[group (mathematics)|group]]. The subgroup generated by a subset ''A'' may also be obtained as the set of all finite products of zero or more elements of ''A'' or their inverses. | |||
* [[Normal subgroup]]s of a group. The normal subgroup generated by a subset ''A'' may also be obtained as the subgroup generated by the elements of ''A'' together with all their [[conjugation (group theory)|conjugates]]. | |||
* [[Submodule]]s of a [[module (algebra)]] or [[subspace]]s of a [[vector space]]. The submodule generated by a subset ''A'' may also be obtained as the set of all finite [[linear combination]]s of elements of ''A''. | |||
The principal example of a topological closure system is the family of [[closed set]]s in a [[topological space]]. The corresponding [[closure (topology)|closure operator]] is denoted <math>\overline A</math>. It may also be obtained as the [[union]] of the set ''A'' with its [[limit point]]s. | |||
Latest revision as of 03:00, 14 February 2010
In mathematics a closure operator is a unary operator or function on subsets of a given set which maps a subset to a containing subset with a particular property.
A closure operator on a set X is a function F on the power set of X, , satisfying:
A topological closure operator satisfies the further property
A closed set for F is one of the sets in the image of F
Closure system
A closure system is the set of closed sets of a closure operator. A closure system is defined as a family of subsets of a set X which contains X and is closed under taking arbitrary intersections:
The closure operator F may be recovered from the closure system as
Examples
In many algebraic structures the set of substructures forms a closure system. The corresponding closure operator is often written and termed the substructure "generated" or "spanned" by A. Notable examples include
- Subsemigroups of a semigroup S. The semigroup generated by a subset A may also be obtained as the set of all finite products of one or more elements of A.
- Subgroups of a group. The subgroup generated by a subset A may also be obtained as the set of all finite products of zero or more elements of A or their inverses.
- Normal subgroups of a group. The normal subgroup generated by a subset A may also be obtained as the subgroup generated by the elements of A together with all their conjugates.
- Submodules of a module (algebra) or subspaces of a vector space. The submodule generated by a subset A may also be obtained as the set of all finite linear combinations of elements of A.
The principal example of a topological closure system is the family of closed sets in a topological space. The corresponding closure operator is denoted . It may also be obtained as the union of the set A with its limit points.
