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Concept
Field of sets
Summary
In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in 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 " 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.
A field of sets is a pair consisting of a set and a family of subsets of called an algebra over that has the following properties:
for all
as an element:
Assuming that (1) holds, this condition (2) is equivalent to:
Any/all of the following equivalent conditions hold:
for all
for all
for all integers and all
for all integers and all
In other words, forms a subalgebra of the power set Boolean algebra of (with the same identity element ).
Many authors refer to itself as a field of sets.
Elements of are called points while elements of are called complexes and are said to be the admissible sets of
A field of sets is called a σ−field of sets and the algebra is called a σ-algebra if the following additional condition (4) is satisfied:
Any/both of the following equivalent conditions hold:
for all
for all
For arbitrary set its power set (or, somewhat pedantically, the pair of this set and its power set) is a field of sets. If is finite (namely, -element), then is finite (namely, -element). It appears that every finite field of sets (it means, with finite, while may be infinite) admits a representation of the form with finite ; it means a function that establishes a one-to-one correspondence between and via : where and (that is, ).
Official source
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