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:
“... there are no chains so galling as the chains of ignorance—no fetters so binding as those that bind the soul, and exclude it from the vast field of useful and scientific knowledge. O, had I received the advantages of early education, my ideas would, ere now, have expanded far and wide; but, alas! I possess nothing but moral capability—no teachings but the teachings of the Holy Spirit.”
—Maria Stewart (1803–1879)
“Frankly, I’d like to see the government get out of war altogether and leave the whole field to private industry.”
—Joseph Heller (b. 1923)
“And weren’t there special cemetery flowers,
That, once grief sets to growing, grief may rest:
The flowers will go on with grief awhile,
And no one seem neglecting or neglected?
A prudent grief will not despise such aids.”
—Robert Frost (1874–1963)