Ideal (set theory)

Template:Short description 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:

1. if $A\in I$ and $B\subseteq A,$ then $B\in I,$ and
2. 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 Template:Em.

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 Template:Em or Template:Em, or simply Template:Em or Template:Em if the ideal $I$ is understood from context. If $I$ is an ideal on $X,$ then a subset of $X$ is said to be Template:Em (or just Template:Em) if it is Template:Em 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 Template:Em.

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 $ℬ$ of subsets of $X$ is a proper ideal on $X$ if and only if its Template:Em in $X,$ which is denoted and defined by $X\setminus ℬ:=\{X\setminus B:B\in ℬ\},$ is a proper filter on $X$ (a filter is Template:Em if it is not equal to $\mathrm{\wp }(X)$). The dual of the power set $\mathrm{\wp }(X)$ is itself; that is, $X\setminus \mathrm{\wp }(X)=\mathrm{\wp }(X).$ Thus a non-empty family $ℬ\subseteq \mathrm{\wp }(X)$ is an ideal on $X$ if and only if its dual $X\setminus ℬ$ is a dual ideal on $X$ (which by definition is either the power set $\mathrm{\wp }(X)$ 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 Template:Em on the natural numbers, denoted ${ℐ}_{1/n},$ 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 Template:Em on the natural numbers, denoted ${𝒵}_0,$ 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 Template:Em is the collection of all sets $A$ of real numbers such that the Lebesgue measure of $A$ is zero.
• The Template:Em is the collection of all meager sets of real numbers.

Ideals on other sets

• If $\lambda$ is an ordinal number of uncountable cofinality, the Template:Em 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 Template:Mvar and Template:Mvar on underlying sets Template:Mvar and Template:Mvar 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 $A\subseteq X\times Y,$ $$A\in I\times J\text{ if and only if }\{x\in X:\{y:⟨x,y⟩\in A\}\in ̸J\}\in I$$ That is, a set lies in the product ideal if only a negligible collection of Template:Mvar-coordinates correspond to a non-negligible slice of Template:Mvar in the Template:Mvar-direction. (Perhaps clearer: A set is Template:Em in the product ideal if positively many Template:Mvar-coordinates correspond to positive slices.)

An ideal Template:Mvar on a set Template:Mvar induces an equivalence relation on $\mathrm{\wp }(X),$ the powerset of Template:Mvar, considering Template:Mvar and Template:Mvar to be equivalent (for $A,B$ subsets of Template:Mvar) if and only if the symmetric difference of Template:Mvar and Template:Mvar is an element of Template:Mvar. The quotient of $\mathrm{\wp }(X)$ by this equivalence relation is a Boolean algebra, denoted $\mathrm{\wp }(X)/I$ (read "P of Template:Mvar mod Template:Mvar").

Template:Anchor To every ideal there is a corresponding filter, called its Template:Em. If Template:Mvar is an ideal on Template:Mvar, then the dual filter of Template:Mvar is the collection of all sets $X\setminus A,$ where Template:Mvar is an element of Template:Mvar. (Here $X\setminus A$ denotes the relative complement of Template:Mvar in Template:Mvar; that is, the collection of all elements of Template:Mvar that are Template:Em in Template:Mvar).

Relationships among ideals

If $I$ and $J$ are ideals on $X$ and $Y$ respectively, $I$ and $J$ are Template:Em 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 $\phi :X\setminus A\to Y\setminus B,$ such that for any subset $C\subseteq X,$ $C\in I$ if and only if the image of $C$ under $\phi \in J.$

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

See also

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References

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