ideal🔊

Pertaining to ideas, or to a given idea.

Existing only in the mind; conceptual, imaginary.

Optimal; being the best possibility.

Perfect, flawless, having no defects.

Teaching or relating to the doctrine of idealism.

(mathematics) Not actually present, but considered as present when limits at infinity are included.

A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.

(a perfect standard of beauty, intellect etc.)A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.

(ring theory) A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.

(lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).{{pedia|Boolean prime ideal theorem#Prime ideal theorems}}

(set theory) 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.

(Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.

(algebra) A subsemigroup with the property that if any semigroup element outside of it is added to any one of its members, the result must lie outside of it.{{cite-web|1=en|year=2004|author={{w|Vaughan Pratt}}|work=Chapter 1 : Lattice Theory|url=http://boole.stanford.edu/cs353/handouts/book1.pdf|publisher=boole.stanford.edu|section=§1.3.5}}