Ideal on a set

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 ${\displaystyle X}$ is an ideal on the powerset ${\displaystyle {\mathcal {P}}(X)}$ partially ordered by inclusion), and by ideals on rings (an ideal on ${\displaystyle X}$ is an ideal on the Boolean ring ${\displaystyle {\mathcal {P}}(X)}$). The notion dual to ideals is filters.