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

1. ${\displaystyle \varnothing \in I,}$
2. if ${\displaystyle A\in I}$ and ${\displaystyle B\subseteq A,}$ then ${\displaystyle B\in I,}$ and
3. if ${\displaystyle A,B\in I}$ then ${\displaystyle A\cup B\in I.}$

Some authors add a fourth condition that ${\displaystyle X}$ itself is not in ${\displaystyle I}$; 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.