Ideal (set theory)

In the mathematical field of set theory, an ideal is a 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 "X", an ideal on "X" is a nonempty subset "I" of the powerset of "X", such that:
# if "A" ∈ "I" and "B" ⊆ "A", then also "B" ∈ "I", and
# if "A","B" ∈ "I", then "A"∪"B" ∈ "I".

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.

Terminology

An element of an ideal "I" is said to be "I-null" or "I-negligible", or simply "null" or "negligible" if the ideal "I" is understood from context. If "I" is an ideal on "X", then a subset of "X" is said to be "I-positive" (or just "positive") if it is "not" an element of "I". The collection of all "I"-positive subsets of "X" is denoted "I"⁺.

Examples of ideals

General examples

*For any set "X" and any arbitrarily chosen subset "B" ⊆ "X", the subsets of "B" form an ideal on "X".
*The finite subsets of any set "X" form an ideal on "X".

Ideals on the natural numbers

*The ideal of all finite sets of natural numbers is denoted Fin.
*The "summable ideal" on the natural numbers, denoted $mathcal\{I\}\_\{1/n\}$, is the collection of all sets "A" of natural numbers such that the sum $sum\_\{nin\; A\}frac\{1\}\{n+1\}$ is finite.
*The "ideal of asymptotically zero-density sets" on the natural numbers, denoted $mathcal\{Z\}\_0$, is the collection of all sets "A" of natural numbers such that the fraction of natural numbers less than "n" that belong to "A", tends to zero as "n" tends to infinity. (That is, the asymptotic density of "A" is zero.)

Ideals on the real numbers

*The "measure ideal" is the collection of all sets "A" of real numbers such that the Lebesgue measure of "A" is zero.
*The "meager ideal" is the collection of all meager sets of real numbers.

Ideals on other sets

*If λ is an ordinal number of uncountable cofinality, the "nonstationary ideal" on λ is the collection of all subsets of λ that are not stationary sets. This ideal has been studied extensively by W. Hugh Woodin.

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",:$Ain\; I\; imes\; Jiff\; \{xin\; X|\{y|langle\; x,y\; anglein\; A\}\; otin\; J\}in\; I$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".)

Relationships among ideals

If "I" and "J" are ideals on "X" and "Y" respectively, "I" and "J" are "Rudin–Keisler isomorphic" 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 "A" and "B", elements of "I" and "J" respectively, and a bijection φ : "X" "A" → "Y" "B", such that for any subset "C" of "X", "C" is in "I" if and only if the image of "C" under φ is in "J".

If "I" and "J" are Rudin–Keisler isomorphic, then "P"("X") / "I" and "P"("Y") / "J" are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called "trivial isomorphisms".

References

*cite book|last=Farah|first=Ilijas|series=Memoirs of the AMS|publisher=American Mathematical Society|year=2000|month=November|title=Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers