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
is a nonempty subset of the powerset of
such that: Some authors add a fourth condition that
; 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.
The dual notion of an ideal is a filter.
An element of an ideal
-negligible, or simply null or negligible if the ideal
is understood from context.
-positive (or just positive) if it is not an element of
is a proper ideal on
Given ideals I and J on underlying sets X and Y respectively, one forms the skew or Fubini product
, an ideal on the Cartesian product
That is, a set lies 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 positive 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
the powerset of X, considering A and B to be equivalent (for
subsets of X) if and only if the symmetric difference of A and B is an element of I.
by this equivalence relation is a Boolean algebra, denoted
(read "P of X mod 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
denotes the relative complement of A in X; that is, the collection of all elements of X that are not in A).
are Rudin–Keisler isomorphic 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
are isomorphic as Boolean algebras.
Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called trivial isomorphisms.