Ideal (set theory)

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 $X,$ an ideal $I$ on $X$ is a nonempty subset of the powerset of $X,$ such that:

if $A\in I$ and $B\subseteq A,$ then $B\in I,$ and
if $A,B\in I$ then $A\cup B\in I.$

Some authors add a fourth condition that $X$ itself is not in $I$ ; ideals with this extra property are called proper ideals.

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 $I$ 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 $I^{+}.$

If $I$ is a proper ideal on $X$ and for every $A\subseteq X$ either $A\in I$ or $X\setminus A\in I,$ then $I$ is a prime ideal.

Examples of ideals

General examples

For any set $X$ and any arbitrarily chosen subset $B\subseteq X,$ the subsets of $B$ form an ideal on $X.$ For finite $X,$ all ideals are of this form.
The finite subsets of any set $X$ form an ideal on $X.$
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 $X$ is an ideal that covers $X.$
A non-empty family ${\mathcal {B}}$ of subsets of $X$ is a proper ideal on $X$ if and only if its dual in $X,$ which is denoted and defined by $X\setminus {\mathcal {B}}:=\{X\setminus B:B\in {\mathcal {B}}\},$ is a proper filter on $X$ (a filter is proper if it is not equal to $\wp (X)$ ). The dual of the power set $\wp (X)$ is itself; that is, $X\setminus \wp (X)=\wp (X).$ Thus a non-empty family ${\mathcal {B}}\subseteq \wp (X)$ is an ideal on $X$ if and only if its dual $X\setminus {\mathcal {B}}$ is a dual ideal on $X$ (which by definition is either the power set $\wp (X)$ or else a proper filter 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 $\sum _{n\in A}{\frac {1}{n+1}}$ is finite. See small set.
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 $\lambda$ is an ordinal number of uncountable cofinality, the nonstationary ideal on $\lambda$ is the collection of all subsets of $\lambda$ 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 skew or Fubini product $I\times J$ , an ideal on the Cartesian product $X\times Y,$ as follows: For any subset $A\subseteq X\times Y,$ $A\in I\times J\quad {\text{ if and only if }}\quad \{x\in X\;:\;\{y:\langle x,y\rangle \in A\}\not \in J\}\in I$ That is, a set lies 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 $\wp (X),$ 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 $\wp (X)$ by this equivalence relation is a Boolean algebra, denoted $\wp (X)/I$ (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 $X\setminus A,$ where $A$ is an element of $I$ . (Here $X\setminus A$ 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 $\varphi :X\setminus A\to Y\setminus B,$ such that for any subset $C\subseteq X,$ $C\in I$ if and only if the image of $C$ under $\varphi \in J.$

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

References

Farah, Ilijas (November 2000). Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers. Memoirs of the AMS. American Mathematical Society. ISBN 9780821821176.