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 {\displaystyle X} is an ideal on the powerset P ( X ) {\displaystyle {\mathcal {P}}(X)} partially ordered by inclusion), and by ideals on rings (an ideal on X {\displaystyle X} is an ideal on the Boolean ring P ( X ) {\displaystyle {\mathcal {P}}(X)} ). The notion dual to ideals is filters.
Definition Given a set X {\displaystyle X} , an ideal I {\displaystyle {\mathcal {I}}} on X {\displaystyle X} is a set of subsets of X {\displaystyle X} such that:
I {\displaystyle {\mathcal {I}}} is downwards-closed: If A , B ⊆ X {\displaystyle A,B\subseteq X} are such that A ∈ I {\displaystyle A\in {\mathcal {I}}} and B ⊆ A {\displaystyle B\subseteq A} then B ∈ I {\displaystyle B\in {\mathcal {I}}} ,
I {\displaystyle {\mathcal {I}}} is closed under finite unions: ∅ ∈ I {\displaystyle \varnothing \in I} , and if A ∈ I {\displaystyle A\in {\mathcal {I}}} and B ∈ I {\displaystyle B\in {\mathcal {I}}} then A ∪ B ∈ I {\displaystyle A\cup B\in {\mathcal {I}}} . A proper ideal is an ideal that is proper as a subset of the powerset P ( X ) {\displaystyle {\mathcal {P}}(X)} . By contrast, P ( X ) {\displaystyle {\mathcal {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 {\displaystyle X} . Some authors adopt the convention that an ideal must be proper by definition.
Terminology An element of an ideal I {\displaystyle I} is said to be I {\displaystyle I} -null or I {\displaystyle I} -negligible, or simply null or negligible if the ideal I {\displaystyle I} is understood from context. If I {\displaystyle I} is an ideal on X , {\displaystyle X,} then a subset of X {\displaystyle X} is said to be I {\displaystyle I} -positive (or just positive) if it is not an element of I . {\displaystyle I.} The collection of all I {\displaystyle I} -positive subsets of X {\displaystyle X} is denoted I + . {\displaystyle I^{+}.}
If I {\displaystyle I} is a proper ideal on X {\displaystyle X} and for every A ⊆ X {\displaystyle A\subseteq X} either A ∈ I {\displaystyle A\in I} or X ∖ A ∈ I , {\displaystyle X\setminus A\in I,} then I {\displaystyle I} is a prime ideal.
Examples of ideals
General examples For any set X {\displaystyle X} and any arbitrarily chosen subset B ⊆ X , {\displaystyle B\subseteq X,} the subsets of B {\displaystyle B} form an ideal on X . {\displaystyle X.} For finite X , {\displaystyle X,} all ideals are of this form. The finite subsets of any set X {\displaystyle X} form an ideal on X . {\displaystyle 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 {\displaystyle X} is an ideal that covers X . {\displaystyle X.}
… excerpt ends here. Continue reading the full article.
