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 .

See also

References

 

Set theory