View source for Ideal (set theory)

{{short description|Non-empty family of sets that is closed under finite unions and subsets}}
In the mathematical field of [[Set theory|set theory]], an '''ideal''' is a partially ordered collection of [[Set (mathematics)|sets]] that are considered to be "small" or "negligible". Every [[Subset|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 (set theory)|union]] of any two elements of the ideal must also be in the ideal.

More formally, given a set <math>X,</math> an ideal <math>I</math> on <math>X</math> is a [[Empty set|nonempty]] subset of the powerset of <math>X,</math> such that:

# <math>\varnothing \in I,</math>
# if <math>A \in I</math> and <math>B \subseteq A,</math> then <math>B \in I,</math> and
# if <math>A, B \in I</math> then <math>A \cup B \in I.</math>

Some authors add a fourth condition that <math>X</math> itself is not in <math>I</math>; ideals with this extra property are called '''{{em|proper ideals}}'''.

Ideals in the set-theoretic sense are exactly [[Ideal (order theory)|ideals in the order-theoretic sense]], where the relevant order is set inclusion. Also, they are exactly [[Ideal (ring theory)|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 (set theory)|filter]].

==Terminology==

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

If <math>I</math> is a proper ideal on <math>X</math> and for every <math>A \subseteq X</math> either <math>A \in I</math> or <math>X \setminus A \in I,</math> then <math>I</math> is a '''{{em|prime ideal}}'''.

==Examples of ideals==

===General examples===

* For any set <math>X</math> and any arbitrarily chosen subset <math>B \subseteq X,</math> the subsets of <math>B</math> form an ideal on <math>X.</math> For finite <math>X,</math> all ideals are of this form.
* The [[Finite set|finite subsets]] of any set <math>X</math> form an ideal on <math>X.</math>
* For any [[Measure space|measure space]], subsets of sets of measure zero.
* For any [[Measure space|measure space]], sets of finite measure. This encompasses finite subsets (using [[Counting measure|counting measure]]) and small sets below.
* A [[Bornology|bornology]] on a set <math>X</math> is an ideal that [[Cover (topology)|covers]] <math>X.</math> 
* A non-empty family <math>\mathcal{B}</math> of subsets of <math>X</math> is a proper ideal on <math>X</math> if and only if its {{em|dual}} in <math>X,</math> which is denoted and defined by <math>X \setminus \mathcal{B} := \{X \setminus B : B \in \mathcal{B}\},</math> is a proper [[Filter (set theory)|filter]] on <math>X</math> (a filter is {{em|proper}} if it is not equal to <math>\wp(X)</math>). The dual of the [[Power set|power set]] <math>\wp(X)</math> is itself; that is, <math>X \setminus \wp(X) = \wp(X).</math> Thus a non-empty family <math>\mathcal{B} \subseteq \wp(X)</math> is an ideal on <math>X</math> if and only if its dual <math>X \setminus \mathcal{B}</math> is a dual ideal on <math>X</math> (which by definition is either the power set <math>\wp(X)</math> or else a proper filter on <math>X</math>).

===Ideals on the natural numbers===

* The ideal of all finite sets of [[Natural number|natural number]]s is denoted Fin.
* The {{em|summable ideal}} on the natural numbers, denoted <math>\mathcal{I}_{1/n},</math> is the collection of all sets <math>A</math> of natural numbers such that the sum <math>\sum_{n\in A}\frac{1}{n+1}</math> is finite. See small set.
* The {{em|ideal of asymptotically zero-density sets}} on the natural numbers, denoted <math>\mathcal{Z}_0,</math> is the collection of all sets <math>A</math> of natural numbers such that the fraction of natural numbers less than <math>n</math> that belong to <math>A,</math> tends to zero as <math>n</math> tends to infinity. (That is, the asymptotic density of <math>A</math> is zero.)

===Ideals on the real numbers===

* The {{em|measure ideal}} is the collection of all sets <math>A</math> of [[Real number|real number]]s such that the [[Lebesgue measure]] of <math>A</math> is zero.
* The {{em|meager ideal}} is the collection of all meager sets of real numbers.

===Ideals on other sets===

* If <math>\lambda</math> is an [[Ordinal number|ordinal number]] of uncountable [[Cofinality|cofinality]], the {{em|nonstationary ideal}} on <math>\lambda</math> is the collection of all subsets of <math>\lambda</math> that are not [[Stationary set|stationary set]]s. This ideal has been studied extensively by [[Biography:W. Hugh Woodin|W. Hugh Woodin]].

==Operations on ideals==

Given ideals {{mvar|I}} and {{mvar|J}} on underlying sets {{mvar|X}} and {{mvar|Y}} respectively, one forms the product <math>I \times J</math> on the [[Cartesian product]] <math>X \times Y,</math> as follows: For any subset <math>A \subseteq X \times Y,</math>
<math display="block">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</math>
That is, a set is negligible in the product ideal if only a negligible collection of {{mvar|x}}-coordinates correspond to a non-negligible slice of {{mvar|A}} in the {{mvar|y}}-direction. (Perhaps clearer: A set is {{em|positive}} in the product ideal if positively many {{mvar|x}}-coordinates correspond to positive slices.)

An ideal {{mvar|I}} on a set {{mvar|X}} induces an [[Equivalence relation|equivalence relation]] on <math>\wp(X),</math> the powerset of {{mvar|X}}, considering {{mvar|A}} and {{mvar|B}} to be equivalent (for <math>A, B</math> subsets of {{mvar|X}}) if and only if the [[Symmetric difference|symmetric difference]] of {{mvar|A}} and {{mvar|B}} is an element of {{mvar|I}}. The [[Quotient set|quotient]] of <math>\wp(X)</math> by this equivalence relation is a [[Boolean algebra (structure)|Boolean algebra]], denoted <math>\wp(X) / I</math> (read "P of {{mvar|X}} mod {{mvar|I}}").

{{anchor|Dual filter}} To every ideal there is a corresponding [[Filter (set theory)|filter]], called its {{em|dual filter}}. If {{mvar|I}} is an ideal on {{mvar|X}}, then the dual filter of {{mvar|I}} is the collection of all sets <math>X \setminus A,</math> where {{mvar|A}} is an element of {{mvar|I}}. (Here <math>X \setminus A</math> denotes the relative complement of {{mvar|A}} in {{mvar|X}}; that is, the collection of all elements of {{mvar|X}} that are {{em|not}} in {{mvar|A}}).

==Relationships among ideals==

If <math>I</math> and <math>J</math> are ideals on <math>X</math> and <math>Y</math> respectively, <math>I</math> and <math>J</math> are {{em|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 <math>A</math> and <math>B,</math> elements of <math>I</math> and <math>J</math> respectively, and a [[Bijection|bijection]] <math>\varphi : X \setminus A \to Y \setminus B,</math> such that for any subset <math>C \subseteq X,</math> <math>C \in I</math> if and only if the [[Image (mathematics)|image]] of <math>C</math> under <math>\varphi \in J.</math>

If <math>I</math> and <math>J</math> are Rudin–Keisler isomorphic, then <math>\wp(X) / I</math> and <math>\wp(Y) / J</math> are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called {{em|trivial isomorphisms}}.

==See also==

* {{annotated link|Bornology}}
* {{annotated link|Filter (mathematics)}}
* {{annotated link|Filter (set theory)}}
* {{annotated link|Ideal (order theory)}}
* {{annotated link|Ideal (ring theory)}}
* {{annotated link|Pi-system|{{pi}}-system}}

==References==

{{reflist}}

* {{cite book|last=Farah|first=Ilijas|series=Memoirs of the AMS|publisher=American Mathematical Society|date=November 2000|title=Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers|isbn=9780821821176|url=https://books.google.com/books?id=IP7TCQAAQBAJ&q=ideal+OR+ideals}}

[[Category:Set theory]]

{{Sourceattribution|Ideal (set theory)}}