View source for Ring of sets

{{Short description|Family closed under unions and relative complements}}
{{distinguish|Ring (mathematics)}}
In [[mathematics]], there are two different notions of a '''ring of sets''', both referring to certain [[Family of sets|families of sets]].

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 [[union (set theory)|union]] and [[intersection (set theory)|intersection]].<ref name="b37">{{citation
 | last = Birkhoff | first = Garrett | authorlink = Garrett Birkhoff
 | doi = 10.1215/S0012-7094-37-00334-X
 | issue = 3
 | journal = Duke Mathematical Journal
 | mr = 1546000
 | pages = 443–454
 | title = Rings of sets
 | volume = 3
 | year = 1937}}.</ref>  That is, the following two statements are true for all sets <math>A</math> and <math>B</math>,
#<math>A,B\in\mathcal{R}</math> implies <math>A \cup B \in \mathcal{R}</math> and
#<math>A,B\in\mathcal{R}</math> implies <math>A \cap B \in \mathcal{R}.</math>

In [[measure theory]], a nonempty family of sets <math>\mathcal{R}</math> is called a ring (of sets) if it is closed under union and [[relative complement]] (set-theoretic difference).<ref>{{citation|title=Measure Theory and Integration|first=Gar|last=De Barra|publisher=Horwood Publishing|year= 2003|isbn=9781904275046|page=13}}.</ref> That is, the following two statements are true for all sets <math>A</math> and <math>B</math>,
#<math>A, B \in \mathcal{R}</math> implies <math>A \cup B \in \mathcal{R}</math> and
#<math>A, B \in \mathcal{R}</math> implies <math>A \setminus B \in \mathcal{R}.</math>
This implies that a ring in the measure-theoretic sense always contains the [[empty set]]. Furthermore, for all sets {{mvar|A}} and {{mvar|B}},
:<math>A\cap B=A\setminus(A\setminus B),</math>
which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic sense.

==Examples==

If {{mvar|X}} is any set, then the [[power set]] of {{mvar|X}} (the family of all subsets of {{mvar|X}}) forms a ring of sets in either sense.

If {{math|(''X'', ≤)}} is a [[partially ordered set]], then its [[upper set]]s (the subsets of {{mvar|X}} with the additional property that if {{mvar|x}} belongs to an upper set ''U'' and {{math|''x'' ≤ ''y''}}, then {{mvar|y}} must also belong to {{mvar|U}}) are closed under both intersections and unions. However, in general it will not be closed under differences of sets.

The [[open set]]s and [[closed set]]s of any [[topological space]] are closed under both unions and intersections.<ref name="b37"/>

On the real line {{math|'''R'''}}, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form {{open-closed|''a'', ''b''}}, with {{math|''a'', ''b'' ∈ '''R'''}} is a ring in the measure-theoretic sense. 

If {{mvar|T}} is any transformation defined on a space, then the sets that are mapped into themselves by {{mvar|T}} are closed under both unions and intersections.<ref name="b37"/>

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"/>

==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"/> 

A family of sets closed under union and relative complement is also closed under [[symmetric difference]] and intersection. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement. This is due to the identities
#<math>A \cup B = (A\, \triangle\, B)\, \triangle\, (A \cap B)</math> and
#<math>A \setminus B = A\, \triangle\, (A \cap B).</math>
Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a [[boolean ring]].

In the measure-theoretic sense, a {{em|[[Sigma-ring|σ-ring]]}} is a ring closed under {{em|countable}} unions, and a [[Delta-ring|δ-ring]] is a ring closed under countable intersections. Explicitly, a σ-ring over <math>X</math> is a set <math>\mathcal{F}</math> such that for any sequence <math>\{A_k\}_{k=1}^\infty \subseteq \mathcal{F},</math> we have <math display=inline> \bigcup_{k=1}^\infty A_k \in \mathcal{F}.</math> 

Given a set <math>X,</math> a {{em|[[field of sets]]}} − also called an {{em|algebra over <math>X</math>}} − is a ring that contains <math>X.</math> This definition entails that an algebra is closed under absolute complement <math>A^c = X \setminus A.</math> A [[σ-algebra]] is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains <math>X.</math> In fact, by [[de Morgan's laws]], a δ-ring that contains <math>X</math> is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of [[Probability theory|probability]] and the definition of [[Measure (mathematics)|measures]].

A '''{{em|semiring (of sets)}}''' is a family of sets <math>\mathcal{S}</math> with the properties
<ol>
<li><math>\varnothing \in \mathcal{S},</math>
* If (3) holds, then <math>\varnothing \in \mathcal{S}</math> if and only if <math>\mathcal{S} \neq \varnothing.</math></li>
<li><math>A, B \in \mathcal{S}</math> implies <math>A \cap B \in \mathcal{S},</math> and</li>
<li><math>A, B \in \mathcal{S}</math> implies <math>A \setminus B = \bigcup_{i=1}^n C_i</math> for some disjoint <math> C_1, \ldots, C_n \in \mathcal{S}.</math></li>
</ol>
Every ring (in the measure theory sense) is a semi-ring. On the other hand, <math>\mathcal{S} := \{\emptyset,\{x\},\{y\}\}</math> on <math>X = \{x,y\}</math> is a semi-ring but not a ring, since it is not closed under unions.

A '''{{em|{{visible anchor|semialgebra}}}}'''{{sfn|Durrett|2019|pp=3-4}} or '''{{em|elementary family}}''' {{sfn|Folland|1999|p=23}} is a collection <math>\mathcal{S}</math> of subsets of <math>X</math> satisfying the semiring properties except with (3) replaced with:
* If <math>E \in \mathcal{S}</math> then there exists a finite number of mutually [[disjoint sets]] <math>C_1, \ldots, C_n \in \mathcal{S}</math> such that <math>X \setminus E = \bigcup_{i=1}^n C_i.</math>

This condition is stronger than (3), which can be seen as follows. If <math>\mathcal{S}</math> is a semialgebra and <math>E, F \in \mathcal{S}</math>, then we can write <math>F^c = F_1 \cup \ldots \cup F_n</math> for disjoint <math>F_i \in S</math>. Then:
<math display="block">E \setminus F = E \cap F^c = E \cap (F_1 \cup \ldots \cup F_n) = (E \cap F_1) \cup \ldots \cup (E \cap F_n)</math>

and every <math>E \cap F_i \in S</math> since it is closed under intersection, and disjoint since they are contained in the disjoint <math>F_i</math>'s. Moreover the condition is ''strictly'' stronger: any <math>S</math> that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set <math>X</math>).

==See also==

* {{annotated link|Algebra of sets}}
* {{annotated link|Delta-ring|{{delta}}-ring}}
* {{annotated link|Field of sets}}
* {{annotated link|Dynkin system|{{lambda}}-system (Dynkin system)}}
* {{annotated link|Monotone class}}
* {{annotated link|Pi-system|{{pi}}-system}}
* {{annotated link|σ-algebra}}
* {{annotated link|Sigma-ideal|{{sigma}}-ideal}}
* {{annotated link|Sigma-ring|{{sigma}}-ring}}

{{Families of sets}}

==References==
{{reflist}}

==Sources==
{{sfn whitelist|CITEREFDurrett2019}}
* {{Durrett Probability Theory and Examples 5th Edition}}
* {{cite book |last=Folland |first=Gerald B. |date=1999 |title=Real Analysis: Modern Techniques and Their Applications |edition=2nd |publisher=John Wiley & Sons |isbn=0-471-31716-0}}

==External links==

* [http://www.encyclopediaofmath.org/index.php/Ring_of_sets Ring of sets] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{DEFAULTSORT:Ring Of Sets}}

[[Category:Families of sets]]
[[Category:Measure theory]]