Wikipedia / Ideal on a set
Wikipedia / 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 partially ordered by inclusion), and by ideals on rings is an ideal on the Boolean ring). The notion dual to ideals is filters.
Definition
, an ideal such that:
is downwards-closed,
,.
A proper ideal itself, consisting of all possible subsets, is called the improper ideal. Some authors adopt the convention that an ideal must be proper by definition.
Terminology
is said to be or , or simply or is said to be (or just ) if it is
is a .
Examples of ideals
General examples
all ideals are of this form.
The finite subsets
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.
if and only if its is a proper filter (a filter is ). The dual of the power set is a dual ideal).
Ideals on the natural numbers
The ideal of all finite sets of natural numbers is denoted Fin.
The is finite. See small set.
The tends to infinity. (That is, the asymptotic density is zero.)
Ideals on the real numbers
The of real numbers such that the Lebesgue measure is zero.
The is the collection of all meager sets of real numbers.
Ideals on other sets
is an ordinal number of uncountable cofinality, the 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, an ideal on the Cartesian product 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 the powerset of , considering and subsets of ) if and only if the symmetric difference of and is an element of . The quotient by this equivalence relation is a Boolean algebra (read "P of mod ").
To every ideal there is a corresponding filter, called its . If is an ideal on , then the dual filter of where is an element of denotes the relative complement of in ; that is, the collection of all elements of that are in ).
Relationships among ideals
are respectively, and a bijection if and only if the image
are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called .