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| {{distinguish|Ring (mathematics)}}
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| In [[mathematics]], there are two different notions of a '''ring of sets''', both referring to certain families of sets.
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| In [[order theory]], a nonempty [[family of sets]] <math>\mathcal{R}</math> is called a '''ring (of sets)''' if it is [[closure (mathematics)|closed]] under [[intersection (set theory)|intersection]] and [[union (set theory)|union]]. That is, for any <math>A,B\in\mathcal{R}</math>,
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| #<math>A \cap B \in \mathcal{R}</math> and
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| #<math>A \cup B \in \mathcal{R}.</math><ref name="b37">{{citation
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| | last = Birkhoff | first = Garrett | authorlink = Garrett Birkhoff
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| | doi = 10.1215/S0012-7094-37-00334-X
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| | issue = 3
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| | journal = Duke Mathematical Journal
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| | mr = 1546000
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| | pages = 443–454
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| | title = Rings of sets
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| | volume = 3
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| | year = 1937}}.</ref>
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| In [[measure theory]], a ring of sets is instead a family closed under unions and [[set-theoretic difference]]s.<ref>{{citation|title=Measure Theory and Integration|first=Gar|last=De Barra|publisher=Horwood Publishing|year= 2003|isbn=9781904275046|page=13}}.</ref> That is, it obeys the two properties
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| #<math>A \setminus B \in \mathcal{R}</math> and
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| #<math>A \cup B \in \mathcal{R}.</math>
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| This implies that it is also closed under intersections, because of the identity
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| :<math>A\cap B=A\setminus(A\setminus B),</math>
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| however, a family of sets that is closed under unions and intersections might not be closed under differences.
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| ==Examples==
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| If ''X'' is any set, then the [[power set]] of ''X'' (the family of all subsets of ''X'') forms a ring of sets in either sense.
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| If (''X'',≤) is a [[partially ordered set]], then its [[upper set]]s (the subsets of ''X'' with the additional property that if ''x'' belongs to an upper set ''U'' and ''x'' ≤ ''y'', then ''y'' must also belong to ''U'') is closed under both intersections and unions. However, in general it will not be closed under differences of sets.
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| The [[open set]]s and [[closed set]]s of any [[topological space]] are closed under both unions and intersections.<ref name="b37"/>
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| If ''T'' is any transformation of a space, then the sets that are mapped into themselves by ''T'' are closed under both unions and intersections.<ref name="b37"/>
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| If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets.<ref name="b37"/>
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| ==Related structures== | |
| A ring of sets (in the order-theoretic sense) forms a [[distributive lattice]] in which the intersection and union operations correspond to the lattice's meet and join operations, respectively. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of [[finite set|finite]] distributive lattices, this is [[Birkhoff's representation theorem]] and the sets may be taken as the lower sets of a partially ordered set.<ref name="b37"/> Every [[field of sets]] and so also any [[σ-algebra]] also is a ring of sets.
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| ==References== | |
| {{reflist}}
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| {{DEFAULTSORT:Ring Of Sets}}
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| [[Category:Set families]]
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