Ideal on a set

Template:Short description 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 (an ideal on a set $X$ is an ideal on the powerset $𝒫(X)$ partially ordered by inclusion), and by ideals on rings (an ideal on $X$ is an ideal on the Boolean ring $𝒫(X)$). The notion dual to ideals is filters.

Definition

Given a set $X$, an ideal $ℐ$ on $X$ is a set of subsets of $X$ such that:

• $ℐ$ is downwards-closed: If $A,B\subseteq X$ are such that $A\in ℐ$ and $B\subseteq A$ then $B\in ℐ$,
• $ℐ$ is closed under finite unions: $\varnothing \in I$,Template:Efn and if $A\in ℐ$ and $B\in ℐ$ then $A\cup B\in ℐ$.

A proper ideal is an ideal that is proper as a subset of the powerset $𝒫(X)$. By contrast, $𝒫(X)$ itself, consisting of all possible subsets, is called the improper ideal. By downwards-closure, an ideal is proper if and only if it does not contain $X$. Some authors adopt the convention that an ideal must be proper by definition.

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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