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 (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 I on X is a nonempty subset of the powerset of X, such that: Some**** authors add a fourth**** condition that**** X itself**** is**** not in**** I;**** ideals**** with**** this**** extra property**** are called**** . 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. The dual notion of an ideal is a filter.

Terminology

An element of an ideal I is said to be or, or simply or if the ideal I is understood from context. If I is an ideal on X, then a subset of X is said to be (or just ) if it is an element of I. The collection of all I-positive subsets of X is denoted I^+. If**** I is**** a proper**** ideal on**** X and for every either**** A *in*** I or**** **** then**** I is**** a .

Examples of ideals

General examples

Ideals on the natural numbers

Ideals on the real numbers

Ideals on other sets

Operations on ideals

Given ideals I and J on underlying sets X and Y respectively, one forms the product I \times J on the Cartesian product X \times Y, as follows: For any subset 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. (Perhaps clearer: A set is in the product ideal if positively many x-coordinates correspond to positive slices.) An ideal I on a set X induces an equivalence relation on \wp(X), the powerset of X, considering A and B to be equivalent (for A, B subsets of X) if and only if the symmetric difference of A and B is an element of I. The quotient of \wp(X) by this equivalence relation is a Boolean algebra, denoted \wp(X) / I (read "P of X mod I"). To every ideal there is a corresponding filter, called its. If I is an ideal on X, then the dual filter of I is the collection of all sets where A is an element of I. (Here denotes the relative complement of A in X; that is, the collection of all elements of X that are in A).

Relationships among ideals

If I and J are ideals on X and Y respectively, I and J are if they are the same ideal except for renaming of the elements of their underlying sets (ignoring negligible sets). More formally, the requirement is that there be sets A and B, elements of I and J respectively, and a bijection such that for any subset C \in I if and only if the image of C under If I and J are Rudin–Keisler isomorphic, then \wp(X) / I and \wp(Y) / J are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called.