In mathematics a field of sets is a pair where is a set and is an algebra over i.e., a non-empty subset of the power set of closed under the intersection and union of pairs of sets and under complements of individual sets. In other words forms a subalgebra of the power set Boolean algebra of . (Many authors refer to itself as a field of sets. The word "field" in "field of sets" is not used with the meaning of field from field theory.) Elements of are called points and those of are called complexes.
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.
Famous quotes containing the words field of, field and/or sets:
“Hardly a book of human worth, be it heavens own secret, is honestly placed before the reader; it is either shunned, given a Periclean funeral oration in a hundred and fifty words, or interred in the potters field of the newspapers back pages.”
—Edward Dahlberg (19001977)
“Frankly, Id like to see the government get out of war altogether and leave the whole field to private industry.”
—Joseph Heller (b. 1923)
“Nothing sets a person up more than having something turn out just the way its supposed to be, like falling into a Swiss snowdrift and seeing a big dog come up with a little cask of brandy round its neck.”
—Claud Cockburn (19041981)