Ideal (set Theory) - Operations On Ideals

Operations On Ideals

Given ideals I and J on underlying sets X and Y respectively, one forms the product I×J on the Cartesian product X×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. (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 P(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 P(X) by this equivalence relation is a Boolean algebra, denoted P(X) / I (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 X \ A, where A is an element of I. (Here X \ A denotes the relative complement of A in X; that is, the collection of all elements of X that are not in A.)