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''closure operation, in a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226401.png" />''
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''closure operation, in a [[partially ordered set]] $M$ ''
  
A single-valued mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226402.png" /> into itself, associating with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226403.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226404.png" />, called the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226405.png" />, in such a way that the following conditions hold: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226406.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226408.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c0226409.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264010.png" /> is said to be closed if it is its own closure. A closure operation in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264011.png" /> is uniquely determined by specifying the system of all closed elements. In the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264012.png" /> is the set of all subsets of an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264013.png" />, ordered by inclusion, one speaks of a closure operation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264014.png" />. On any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264015.png" />, a closure operation can be defined by taking the closed subsets to be any system of subsets that includes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264016.png" /> itself and is closed under arbitrary intersections. Two partially ordered sets with closure operations are said to be isomorphic if there is an isomorphism of the partially ordered sets under which the images and pre-images of closed sets are closed. Considerable importance is given in mathematics to closure operations on the set of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264017.png" /> which satisfy the following additional assumption: The empty set is closed, and the closure of the union of two subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264018.png" /> equals the union of their closures. A closure operation satisfying this condition is called a topology on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264019.png" />.
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A single-valued mapping of $M$ into itself, associating with each element $a \in M$ an element $\bar a \in M$, called the closure of $a$, in such a way that the following conditions hold: 1) $a \le \bar a$; 2) if $a \le b$, then $\bar a \le \bar b$; and 3) $\overline{\bar a} = \bar a$. An element $a \in M$ is said to be closed if it is its own closure. A closure operation in a set $M$ is uniquely determined by specifying the system of all closed elements.  
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In the particular case when $M$ is the set of all subsets of an arbitrary set $X$, ordered by inclusion, one speaks of a closure operation on $X$. On any set $X$, a closure operation can be defined by taking the closed subsets to be any system of subsets that includes $X$ itself and is closed under arbitrary intersections. Two partially ordered sets with closure operations are said to be isomorphic if there is an isomorphism of the partially ordered sets under which the images and pre-images of closed sets are closed.  
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Considerable importance is given in mathematics to closure operations on the set of all subsets of $X$ which satisfy the following additional assumption: The empty set is closed, and the closure of the union of two subsets of $X$ equals the union of their closures. A closure operation satisfying this condition is called a topology on the set $X$: cf. [[Topological structure (topology)]]
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
 
  
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[1] P.M. Cohn, "Universal algebra", Reidel (1981) {{MR|0620952}} {{ZBL|0461.08001}}
  
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[2] A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}}
  
 
====Comments====
 
====Comments====
Instead of closure operation the terms closure operator and join operator are also used (cf., e.g., [[#References|[1]]]). A family of subsets of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264020.png" /> is called a closure system if it is closed under arbitrary intersections. An example is the collection of all subspaces of an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264021.png" />, and as explained above this induces a closure operation on the subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264022.png" />.
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Instead of closure operation the terms closure operator and join operator are also used (cf., e.g., [[#References|[1]]]). A family of subsets of a given set $X$ is called a closure system if it is closed under arbitrary intersections. An example is the collection of all subspaces of an [[affine space]] $A$, and as explained above this induces a closure operation on the subsets of $A$.
  
Closure operations commuting with finite unions are often called Kuratowskian closure operations, in honour of [[#References|[a1]]]. A [[Boolean algebra|Boolean algebra]] equipped with a closure operation is sometimes called a closure algebra (see [[#References|[a2]]]).
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Closure operations commuting with finite unions are often called [[Kuratowski closure operator]]s, in honour of [[#References|[a1]]]. A [[Boolean algebra]] equipped with a closure operation is sometimes called a closure algebra (see [[#References|[a2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,  "Sur l'opération <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022640/c02264023.png" /> de l'analysis situs"  ''Fund. Math.'' , '''3'''  (1922)  pp. 182–199</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C.C. McKinsey,  A. Tarski,  "On closed elements in closure algebras"  ''Ann. Math. (2)'' , '''47'''  (1946)  pp. 122–162</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,  "Sur l'opération $\bar A$ de l'analysis situs"  ''Fund. Math.'' , '''3'''  (1922)  pp. 182–199</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C.C. McKinsey,  A. Tarski,  "On closed elements in closure algebras"  ''Ann. Math. (2)'' , '''47'''  (1946)  pp. 122–162</TD></TR>
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</table>
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{{TEX|done}}
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[[Category:Order, lattices, ordered algebraic structures]]
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[[Category:General topology]]

Latest revision as of 00:07, 15 February 2024

closure operation, in a partially ordered set $M$

A single-valued mapping of $M$ into itself, associating with each element $a \in M$ an element $\bar a \in M$, called the closure of $a$, in such a way that the following conditions hold: 1) $a \le \bar a$; 2) if $a \le b$, then $\bar a \le \bar b$; and 3) $\overline{\bar a} = \bar a$. An element $a \in M$ is said to be closed if it is its own closure. A closure operation in a set $M$ is uniquely determined by specifying the system of all closed elements.

In the particular case when $M$ is the set of all subsets of an arbitrary set $X$, ordered by inclusion, one speaks of a closure operation on $X$. On any set $X$, a closure operation can be defined by taking the closed subsets to be any system of subsets that includes $X$ itself and is closed under arbitrary intersections. Two partially ordered sets with closure operations are said to be isomorphic if there is an isomorphism of the partially ordered sets under which the images and pre-images of closed sets are closed.

Considerable importance is given in mathematics to closure operations on the set of all subsets of $X$ which satisfy the following additional assumption: The empty set is closed, and the closure of the union of two subsets of $X$ equals the union of their closures. A closure operation satisfying this condition is called a topology on the set $X$: cf. Topological structure (topology)

References

[1] P.M. Cohn, "Universal algebra", Reidel (1981) MR0620952 Zbl 0461.08001

[2] A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901

Comments

Instead of closure operation the terms closure operator and join operator are also used (cf., e.g., [1]). A family of subsets of a given set $X$ is called a closure system if it is closed under arbitrary intersections. An example is the collection of all subspaces of an affine space $A$, and as explained above this induces a closure operation on the subsets of $A$.

Closure operations commuting with finite unions are often called Kuratowski closure operators, in honour of [a1]. A Boolean algebra equipped with a closure operation is sometimes called a closure algebra (see [a2]).

References

[a1] C. Kuratowski, "Sur l'opération $\bar A$ de l'analysis situs" Fund. Math. , 3 (1922) pp. 182–199
[a2] J.C.C. McKinsey, A. Tarski, "On closed elements in closure algebras" Ann. Math. (2) , 47 (1946) pp. 122–162
How to Cite This Entry:
Closure relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_relation&oldid=11785
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article