In the mathematical field of set theory, an ideal is a partially ordered 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, and the union of any two elements of the ideal must also be in the ideal. More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that:
An element of an ideal I is said to be I-null or I-negligible, or simply null or negligible if the ideal I is understood from context. If I is an ideal on X, then a subset of X is said to be I-positive if it is not an element of I. The collection of all I-positive subsets of X is denoted I+. If is a proper ideal on and for every either or, then I is a prime ideal.
Examples of ideals
General examples
For any set X and any arbitrarily chosen subset B ⊆ X, the subsets of B form an ideal on X. For finite X, all ideals are of this form.
The ideal of all finite sets of natural numbers is denoted Fin.
The summable ideal on the natural numbers, denoted, is the collection of all sets A of natural numbers such that the sum is finite. Seesmall set.
The ideal of asymptotically zero-density sets on the natural numbers, denoted, is the collection of all sets A of natural numbers such that the fraction of natural numbers less than n that belong to A, tends to zero as n tends to infinity.
The measure ideal is the collection of all sets A of real numbers such that the Lebesgue measure of A is zero.
The meager ideal is the collection of all meager sets of real numbers.
Ideals on other sets
If λ is an ordinal number of uncountable cofinality, the nonstationary ideal on λ is the collection of all subsets of λ that are not stationary sets. This ideal has been studied extensively by W. Hugh Woodin.
Operations on ideals
Given ideals I and J on underlying sets X and Y respectively, one forms the product I×J on the Cartesian productX×Y, as follows: For any subset A ⊆ X×Y, That is, a set is negligible in the product ideal if only a negligible collection of x-coordinates correspond to a non-negligible slice of A in the y-direction. An ideal I on a set X induces an equivalence relation on P, the powerset of X, considering A and B to be equivalent if and only if the symmetric difference of A and B is an element of I. The quotient of P by this equivalence relation is a Boolean algebra, denoted P / I. To every ideal there is a corresponding filter, called its dual filter. If I is an ideal on X, then the dual filter of I is the collection of all sets X \ A, where A is an element of I.
Relationships among ideals
If I and J are ideals on X and Y respectively, I and J are Rudin–Keisler isomorphic if they are the same ideal except for renaming of the elements of their underlying sets. More formally, the requirement is that there be sets A and B, elements of I and J respectively, and a bijection φ : X \ A → Y \ B, such that for any subset C of X, C is in I if and only if the image ofC under φ is in J. If I and J are Rudin–Keisler isomorphic, then P / I and P / J are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called trivial isomorphisms.
