Field of sets

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a field of sets is a <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a> consisting of a pair 
 
 
 
 (
 X
 ,
 
 
 F
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {F}})}
 
 consisting of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and a <a href="/facts/Family_of_sets/WbYof7lP">family</a> 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 of <a href="/facts/Subset/SJGERmbA">subsets</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 called an algebra over 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that contains the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> as an element, and is closed under the operations of taking <a href="/facts/Complement_(set_theory)/yWUePIYY">complements</a> in 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 finite <a href="/facts/Union_(set_theory)/KVVXKfWl">unions</a>, and finite <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersections</a>.
Fields of sets should not be confused with <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> in <a href="/facts/Ring_theory/VW6svEFe">ring theory</a> nor with <a href="/facts/Field_(physics)/FJdAKeRA">fields in physics</a>. Similarly the term "algebra over 
 
 
 
 X
 
 
 {\displaystyle X}
 
" is used in the sense of a Boolean algebra and should not be confused with <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebras over fields or rings</a> in ring theory.
Fields of sets play an essential role in the <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> of Boolean algebras. Every Boolean algebra can be represented as a field of sets.

Field of sets open-in-new

Field of sets