ideals

noun

1. A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
2. A subring closed under multiplication by its containing ring.
3. (lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
4. A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.