Ideal (set Theory)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
More formally, given a set X, an ideal on X is a nonempty subset I of the powerset of X, such that:
- if A ∈ I and B ⊆ A, then B ∈ I, and
- if A,B ∈ I, then A∪B ∈ I.
Some authors add a third condition that X itself is not in I; ideals with this extra property are called proper ideals.
Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.
Read more about Ideal (set Theory): Terminology, Operations On Ideals, Relationships Among Ideals
Famous quotes containing the word ideal:
“The ideal of brotherhood of man, the building of the Just City, is one that cannot be discarded without lifelong feelings of disappointment and loss. But, if we are to live in the real world, discard it we must. Its very nobility makes the results of its breakdown doubly horrifying, and it breaks down, as it always will, not by some external agency but because it cannot work.”
—Kingsley Amis (1922–1995)