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 an ideal on is a nonempty subset of the powerset of such that:
if and then and
if then
Some authors add a fourth condition that itself is not in ; 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 is said to be or , or simply or if the ideal is understood from context. If is an ideal on then a subset of is said to be (or just ) if it is an element of The collection of all -positive subsets of is denoted
If is a proper ideal on and for every either or then is a .
Examples of ideals
General examples
For any set and any arbitrarily chosen subset the subsets of form an ideal on For finite all ideals are of this form. The finite subsets of any set form an ideal on For any measure space, subsets of sets of measure zero. For any measure space, sets of finite measure. This encompasses finite subsets (using counting measure) and small sets below. A bornology on a set is an ideal that covers A non-empty family of subsets of is a proper ideal on if and only if its in which is denoted and defined by is a proper filter on (a filter is if it is not equal to ). The dual of the power set is itself; that is, Thus a non-empty family is an ideal on if and only if its dual is a dual ideal on (which by definition is either the power set or else a proper filter on ).
Ideals on the natural numbers
The ideal of all finite sets of natural numbers is denoted Fin. The on the natural numbers, denoted is the collection of all sets of natural numbers such that the sum is finite. See small set. The on the natural numbers, denoted is the collection of all sets of natural numbers such that the fraction of natural numbers less than that belong to tends to zero as tends to infinity. (That is, the asymptotic density of is zero.)
Ideals on the real numbers
The is the collection of all sets of real numbers such that the Lebesgue measure of is zero. The is the collection of all meager sets of real numbers.
Ideals on other sets
If is an ordinal number of uncountable cofinality, the on is the collection of all subsets of that are not stationary sets. This ideal has been studied extensively by W. Hugh Woodin.
Operations on ideals
Given ideals and on underlying sets and respectively, one forms the product on the Cartesian product as follows: For any subset
That is, a set is negligible in the product ideal if only a negligible collection of -coordinates correspond to a non-negligible slice of in the -direction. (Perhaps clearer: A set is in the product ideal if positively many -coordinates correspond to positive slices.)
An ideal on a set induces an equivalence relation on the powerset of , considering and to be equivalent (for subsets of ) if and only if the symmetric difference of and is an element of . The quotient of by this equivalence relation is a Boolean algebra, denoted (read "P of mod ").
To every ideal there is a corresponding filter, called its . If is an ideal on , then the dual filter of is the collection of all sets where is an element of . (Here denotes the relative complement of in ; that is, the collection of all elements of that are in ).
Relationships among ideals
If and are ideals on and respectively, and 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 and elements of and respectively, and a bijection such that for any subset if and only if the image of under
If and are Rudin–Keisler isomorphic, then and are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called .