Ideal (set theory)

In the mathematical field of set theory, an ideal is a 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. ${\displaystyle \emptyset \in I}$

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

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