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.)
Read more about this topic: Ideal (set Theory)
Famous quotes containing the words operations and/or ideals:
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—Jean Scott Rogers. Robert Day. Mr. Blount (Frank Pettingell)
“The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.”
—Calvin Coolidge (1872–1933)