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:
“I see a girl dragged by the wrists
Across a dazzling field of snow,
And there is nothing in me that resists.
Once it would not be so....”
—Philip Larkin (1922–1986)
“Father calls me William, sister calls me Will,
Mother calls me Willie, but the fellers call me Bill!”
—Eugene Field (1850–1895)
“Willing sets you free: that is the true doctrine of will and freedom—thus Zarathustra instructs you.”
—Friedrich Nietzsche (1844–1900)