ideal
UK /aɪˈdɪəl/ · US /aɪˈdi(ə)l/ adjective, noun, proper noun
Forms ideals
- adjective Pertaining to ideas, or to a given idea.
- adjective Existing only in the mind; conceptual, imaginary.
- adjective Optimal; being the best possibility.
- adjective Perfect, flawless, having no defects.
- adjective Teaching or relating to the doctrine of idealism. “the ideal theory or philosophy”
- adjective (mathematics) Not actually present, but considered as present when limits at infinity are included. “ideal point”
- noun A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
- noun (algebra, 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. “Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.”
- noun (algebra, order theory, lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
- noun (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.
- noun (algebra, 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.
- noun (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.
- proper noun A city in Georgia, United States.
- proper noun An unincorporated community in Illinois.
- proper noun An unincorporated community in South Dakota.
Synonyms
conceptual notional imaginary best optimal optimum Lie ideal order ideal paragon nonpareil saint apotheosis nonesuch nonsuch idealistic
Antonyms
Etymology
From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.
Webster’s 1913
- adjective Existing in idea or thought; conceptional; intellectual; mental; as, ideal knowledge.
- adjective Reaching an imaginary standard of excellence; fit for a model; faultless; as, ideal beauty. Byron.
- noun A mental conception regarded as a standard of perfection; a model of excellence, beauty, etc. The ideal is to be attained by selecting and assembling in one whole the beauties and perfections which are usually seen in different individuals, excluding everything defective or unseemly, so as to form a type or model of the species. Thus, the Apollo Belvedere is the ideal of the beauty and proportion of the human frame. Fleming.
Etymology [L. idealis: cf. F. idéal.]
Sources: Wiktionary · WordNet 3.0 · Webster’s 1913 — from the Wiktionary entry “ideal”, CC BY-SA 4.0. Details