Ideal on a set

In mathematics, an ideal on a set is a family of subsets that is closed under subsets and finite unions. Informally, sets that belong to the ideal are considered "small" or "negligible".

The concept is generalized both by ideals on a partially ordered set (an ideal on a set is an ideal on the powerset partially ordered by inclusion), and by ideals on rings (an ideal on is an ideal on the Boolean ring ). The notion dual to ideals is filters.

Definition

Given a set , an ideal on is a set of subsets of such that:

• is downwards-closed: If are such that and then ,
• is closed under finite unions: , and if and then .

A proper ideal is an ideal that is proper as a subset of the powerset (i.e., the only improper ideal is , consisting of all possible subsets). By downwards-closure, an ideal is proper if and only if it does not contain . Some authors adopt the convention that an ideal must be proper by definition.

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 skew or Fubini product , an ideal on the Cartesian product as follows: For any subset

That is, a set lies 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 .

See also

Notes

References