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)
“Vigil strange I kept on the field one night;
When you my son and my comrade dropt at my side that day,
One look I but gave which your dear eyes return’d with a look I
shall never forget,
One touch of your hand to mine O boy, reach’d up as you lay on the ground,”
—Walt Whitman (1819–1892)
“Whether changes in the sibling relationship during adolescence create long-term rifts that spill over into adulthood depends upon the ability of brothers and sisters to constantly redefine their connection. Siblings either learn to accept one another as independent individuals with their own sets of values and behaviors or cling to the shadow of the brother and sister they once knew.”
—Jane Mersky Leder (20th century)