Field of sets

In mathematics, a field of sets is a mathematical structure consisting of a pair ${\displaystyle (X,{\mathcal {F}})}$ consisting of a set ${\displaystyle X}$ and a family ${\displaystyle {\mathcal {F}}}$ of subsets of ${\displaystyle X}$ called an algebra over ${\displaystyle X}$ that contains the empty set as an element, and is closed under the operations of taking complements in ${\displaystyle X,}$ finite unions, and finite intersections.

Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over ${\displaystyle X}$" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.

Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.