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ideal

From Wiktionary, the free dictionary
See also: Ideal, ideał, ideál, and idéal

English

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English Wikipedia has an article on:
Wikipedia

Etymology

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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.

Pronunciation

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  • (UK) IPA(key): /aɪˈdɪəl/
  • (US) IPA(key): /aɪˈdi(ə)l/
    • Audio (US):(file)
  • Rhymes: -iːəl

Adjective

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ideal (comparative more ideal, superlative most ideal)

  1. Pertaining to ideas, or to a given idea.
  2. Existing only in the mind; conceptual, imaginary.
    • 1796, Matthew Lewis, The Monk, Folio Society, published 1985, page 256:
      The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
    • 1816 June – 1817 April/May (date written), [Mary Shelley], “Chapter 4”, in Frankenstein; or, The Modern Prometheus. […], volume (please specify |volume=I to III), London: […] [Macdonald and Son] for Lackington, Hughes, Harding, Mavor, & Jones, published 1 January 1818, →OCLC:
      Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
    • 1836 March – 1837 October, Charles Dickens, “(please specify the chapter name)”, in The Posthumous Papers of the Pickwick Club, London: Chapman and Hall, […], published 1837, →OCLC:
      At first, he began to doubt the reality of his adventures, but the acute pain in his shoulders when he attempted to rise, assured him that the kicking of the goblins was certainly not ideal.
  3. Optimal; being the best possibility.
  4. Perfect, flawless, having no defects.
    • 1751 April 13, Samuel Johnson, The Rambler, Number 112, reprinted in 1825, The Works of Samuel Johnson, LL. D., Volume 1, Jones & Company, page 194,
      There will always be a wide interval between practical and ideal excellence; […] .
  5. Teaching or relating to the doctrine of idealism.
    the ideal theory or philosophy
  6. (mathematics) Not actually present, but considered as present when limits at infinity are included.
    ideal point
    An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

Synonyms

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Derived terms

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Translations

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The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.
Translations to be checked

Noun

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ideal (plural ideals)

  2. A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
    • 1901 [1878], chapter 5, in Constance Garnett, transl., Anna Karenina‎[3], translation of Анна Каренина (Anna Karenina) by Leo Tolstoy, part 6:
      Varvara Andreevna, when I was very young, I set before myself the ideal of the woman I loved and should be happy to call my wife.
    • 1945 April 16, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S.‎[4], Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162:
      With great humility, I call upon all Americans to help me keep our nation united in defense of those ideals which have been so eloquently proclaimed by Franklin Roosevelt. I want in turn to assure my fellow Americans and all of those who love peace and liberty throughout the world that I will support and defend those ideals with all my strength and all my heart.
  3. (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 be the ring of integers and let be its ideal of even integers. Then the quotient ring is a Boolean ring.
    The product of two ideals and is an ideal which is a subset of the intersection of and . This should help to understand why maximal ideals are prime ideals. Likewise, the union of and is a subset of .
    • 2004, K. R. Goodearl; R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:
      In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.
    • 2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45:
      If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.
    • 2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69:
      However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.
  4. (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).[1]
    • 1992, T. S. Fofanova, “General Theory of Lattices”, in Unnamed, transl., Ordered Sets and Lattices II, American Mathematical Society, page 119:
      An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.
    • 2011, George Grätzer, Lattice Theory: Foundation‎[5], Springer (Birkhäuser), page 125:
      1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
    • 2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186:
      Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.
  5. (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.
    Formally, an ideal of a given set is a nonempty subset of the powerset such that: , and .
  6. (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.
    • 1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13:
      If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
    • 2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov, editors, Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3:
      What really put primitive ideals in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such ideal is the annihilator of a very special kind of simple module, namely a highest weight module.
    • 2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73:
      Next let be an arbitrary semisimple Lie algebra. Then can be written uniquely as a direct sum of simple ideals (Theorem 5.2).
  7. (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.[2]
    The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.

Synonyms

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Antonyms

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  • (antonym(s) of “order theory”): filter

Hyponyms

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Derived terms

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Translations

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References

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  1. ^ Boolean prime ideal theorem on Wikipedia.Wikipedia
  2. ^ Vaughan Pratt (2004), Chapter 1 : Lattice Theory‎[1], boole.stanford.edu, §1.3.5

Further reading

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Anagrams

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Asturian

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Etymology

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From Latin ideālis.

Pronunciation

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  • IPA(key): /ideˈal/ [i.ð̞eˈal]
  • Rhymes: -al
  • Syllabification: i‧de‧al

Adjective

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ideal m or f by sense (ideales)

  1. ideal

Noun

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ideal m (plural ideales)

  1. ideal

Further reading

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Catalan

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Etymology

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    Borrowed from Latin ideālis.

    Pronunciation

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    Adjective

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    ideal m or f (masculine and feminine plural ideals)

    1. ideal

    Derived terms

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    Noun

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    ideal m (plural ideals)

    1. ideal

    Chavacano

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    Etymology

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    Inherited from Spanish ideal (“ideal”).

    Pronunciation

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    • Hyphenation: i‧de‧al

    Adjective

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    ideal

    1. ideal
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    Danish

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    Etymology 1

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    From Late Latin ideālis, from Latin idea.

    Adjective

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    ideal (neuter idealt, plural and definite singular attributive ideale)

    1. ideal
      Synonyms: perfekt, fuldkommen
      Coordinate term: ideel

    Inflection

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    Inflection of ideal
    positive comparative superlative
    indefinite common singular ideal mere ideal mest ideal2
    indefinite neuter singular idealt mere ideal mest ideal2
    plural ideale mere ideal mest ideal2
    definite attributive1 ideale mere ideal mest ideale

    1 When an adjective is applied predicatively to something definite,
    the corresponding "indefinite" form is used.
    2 The "indefinite" superlatives may not be used attributively.

    Derived terms

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    Etymology 2

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    A nominalization of the adjective above.

    Noun

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    ideal n (singular definite idealet, plural indefinite idealer)

    1. ideal (quality)
    2. role model, exemplar (person or thing)
      Synonym: forbillede

    Declension

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    Declension of ideal
    neuter
    gender     		singular plural
    indefinite definite indefinite definite
    nominative ideal idealet idealer idealerne
    genitive ideals idealets idealers idealernes

    Derived terms

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    Further reading

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    Galician

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    Etymology

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    From Latin ideālis.

    Pronunciation

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    • IPA(key): /idɛˈal/ [i.ð̞ɛˈɑɫ]
    • Rhymes: -al
    • Hyphenation: i‧de‧al

    Adjective

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    ideal m or f (plural ideais)

    1. ideal

    Derived terms

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    Noun

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    ideal m (plural ideais)

    1. ideal

    Further reading

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    German

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    Etymology

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    Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.

    Pronunciation

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    Adjective

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    ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten)

    1. ideal (optimal, perfect)

    Declension

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    Positive forms of ideal
    number & gender singular plural
    masculine feminine neuter
    predicative er ist ideal sie ist ideal es ist ideal sie sind ideal
    strong declension
    (without article)     		nominative idealer ideale ideales ideale
    genitive idealen idealer idealen idealer
    dative idealem idealer idealem idealen
    accusative idealen ideale ideales ideale
    weak declension
    (with definite article)     		nominative der ideale die ideale das ideale die idealen
    genitive des idealen der idealen des idealen der idealen
    dative dem idealen der idealen dem idealen den idealen
    accusative den idealen die ideale das ideale die idealen
    mixed declension
    (with indefinite article)     		nominative ein idealer eine ideale ein ideales (keine) idealen
    genitive eines idealen einer idealen eines idealen (keiner) idealen
    dative einem idealen einer idealen einem idealen (keinen) idealen
    accusative einen idealen eine ideale ein ideales (keine) idealen
    Comparative forms of ideal
    number & gender singular plural
    masculine feminine neuter
    predicative er ist idealer sie ist idealer es ist idealer sie sind idealer
    strong declension
    (without article)     		nominative idealerer idealere idealeres idealere
    genitive idealeren idealerer idealeren idealerer
    dative idealerem idealerer idealerem idealeren
    accusative idealeren idealere idealeres idealere
    weak declension
    (with definite article)     		nominative der idealere die idealere das idealere die idealeren
    genitive des idealeren der idealeren des idealeren der idealeren
    dative dem idealeren der idealeren dem idealeren den idealeren
    accusative den idealeren die idealere das idealere die idealeren
    mixed declension
    (with indefinite article)     		nominative ein idealerer eine idealere ein idealeres (keine) idealeren
    genitive eines idealeren einer idealeren eines idealeren (keiner) idealeren
    dative einem idealeren einer idealeren einem idealeren (keinen) idealeren
    accusative einen idealeren eine idealere ein idealeres (keine) idealeren
    Superlative forms of ideal
    number & gender singular plural
    masculine feminine neuter
    predicative er ist am idealsten sie ist am idealsten es ist am idealsten sie sind am idealsten
    strong declension
    (without article)     		nominative idealster idealste idealstes idealste
    genitive idealsten idealster idealsten idealster
    dative idealstem idealster idealstem idealsten
    accusative idealsten idealste idealstes idealste
    weak declension
    (with definite article)     		nominative der idealste die idealste das idealste die idealsten
    genitive des idealsten der idealsten des idealsten der idealsten
    dative dem idealsten der idealsten dem idealsten den idealsten
    accusative den idealsten die idealste das idealste die idealsten
    mixed declension
    (with indefinite article)     		nominative ein idealster eine idealste ein idealstes (keine) idealsten
    genitive eines idealsten einer idealsten eines idealsten (keiner) idealsten
    dative einem idealsten einer idealsten einem idealsten (keinen) idealsten
    accusative einen idealsten eine idealste ein idealstes (keine) idealsten

    Derived terms

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    Further reading

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    • “ideal” in Duden online
    • “ideal”, in Digitales Wörterbuch der deutschen Sprache‎[6] (in German)

    Indonesian

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    Indonesian Wikipedia has an article on:
    Wikipedia id

    Etymology

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    From Dutch ideaal, from French idéal, from Medieval Latin ideālis.

    Pronunciation

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    Adjective

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    idéal (comparative lebih ideal, superlative paling ideal)

    1. ideal; optimal, being the best possibility
    2. ideal; pertaining to ideas, or to a given idea

    Noun

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    idéal (plural ideal-ideal)

    1. (mathematics) ideal; a subring closed under multiplication by its containing ring

    Alternative forms

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    Derived terms

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    Further reading

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    Luxembourgish

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    Pronunciation

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    • IPA(key): /ideˈaːl/, [idəˈaːl], /idiˈaːl/

    Adjective

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    ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)

    1. ideal

    Declension

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    Declension of ideal
    singular plural
    masculine feminine neuter
    predicative hien ass ideal si ass ideal et ass ideal si si(nn) ideal
    nominative /
    accusative     		attributive and/or after determiner idealen ideal ideaalt ideal
    independent without determiner ideales idealer
    dative after any declined word idealen idealer idealen idealen
    as first declined word idealem idealem

    Norwegian Bokmål

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    Etymology

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    From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).

    Noun

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    ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene)

    1. ideal

    References

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    Norwegian Nynorsk

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    Etymology

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    From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).

    Noun

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    ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)

    1. ideal

    References

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    Piedmontese

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    Pronunciation

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    Adjective

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    ideal

    1. ideal

    Noun

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    ideal m

    1. ideal

    Portuguese

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    Etymology

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    Borrowed from Latin ideālis, equivalent to ideia +‎ -al.

    Pronunciation

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    • Rhymes: -al, -aw
    • Hyphenation: i‧de‧al

    Adjective

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    ideal m or f (plural ideais)

    1. ideal
    2. notional

    Derived terms

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    Noun

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    ideal m (plural ideais)

    1. ideal
    2. fantasy

    Further reading

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    Romanian

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    Etymology

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    Borrowed from French idéal, from Latin idealis.

    Pronunciation

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    Adjective

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    ideal m or n (feminine singular ideală, masculine plural ideali, feminine/neuter plural ideale)

    1. ideal

    Declension

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    Declension of ideal
    singular plural
    masculine neuter feminine masculine neuter feminine
    nominative-
    accusative     		indefinite ideal ideală ideali ideale
    definite idealul ideala idealii idealele
    genitive-
    dative     		indefinite ideal ideale ideali ideale
    definite idealului idealei idealilor idealelor
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    Serbo-Croatian

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    Pronunciation

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    • IPA(key): /iděaːl/
    • Hyphenation: i‧de‧al

    Noun

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    idèāl m inan (Cyrillic spelling идѐа̄л)

    1. ideal

    Declension

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    Declension of ideal
    singular plural
    nominative idèāl ideali
    genitive ideála ideala
    dative idealu idealima
    accusative ideal ideale
    vocative ideale ideali
    locative idealu idealima
    instrumental idealom idealima

    Spanish

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    Etymology

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    From Latin ideālis.

    Pronunciation

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    • IPA(key): /ideˈal/ [i.ð̞eˈal]
    • Rhymes: -al
    • Syllabification: i‧de‧al

    Adjective

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    ideal m or f (masculine and feminine plural ideales)

    1. ideal

    Derived terms

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    Noun

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    ideal m (plural ideales)

    1. ideal

    Further reading

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    Swedish

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    Pronunciation

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    Adjective

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    ideal (not comparable)

    1. ideal (perfect)
      Synonym: idealisk
      under ideala förhållanden
      under ideal conditions

    Declension

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    Inflection of ideal
    Indefinite positive comparative superlative1
    common singular ideal — —
    neuter singular idealt — —
    plural ideala — —
    masculine plural2 ideale — —
    Definite positive comparative superlative
    masculine singular3 ideale — —
    all ideala — —

    1 The indefinite superlative forms are only used in the predicative.
    2 Dated or archaic.
    3 Only used, optionally, to refer to things whose natural gender is masculine.

    Noun

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    ideal n

    1. ideal; perfect standard
    2. (mathematics) ideal; special subsets of a ring

    Declension

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    Declension of ideal
    nominative genitive
    singular indefinite ideal ideals
    definite idealet idealets
    plural indefinite ideal ideals
    definite idealen idealens
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    References

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    Anagrams

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    Tagalog

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    Alternative forms

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    Etymology

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      Borrowed from Spanish ideal.

      Pronunciation

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      Noun

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      ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

      1. ideal (model to be imitated)
        Synonyms: huwaran, uliran
      2. ambition
        Synonyms: mithi, mithiin, pangarap, ambisyon
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      Adjective

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      ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

      1. ideal (just as what one wants)
      2. worthy of imitation
        Synonym: uliran

      Turkish

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      Etymology

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      Borrowed from French idéal.

      Pronunciation

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      • IPA(key): /ideˈɑl/
      • Hyphenation: i‧de‧al

      Adjective

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      ideal

      1. ideal

      Noun

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      ideal (definite accusative ideali, plural idealler)

      1. ideal

      Declension

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      Declension of ideal
      singular plural
      nominative ideal idealler
      definite accusative ideali idealleri
      dative ideale ideallere
      locative idealde ideallerde
      ablative idealden ideallerden
      genitive idealin ideallerin
      Possessive forms
      nominative
      singular plural
      1st singular idealim ideallerim
      2nd singular idealin ideallerin
      3rd singular ideali idealleri
      1st plural idealimiz ideallerimiz
      2nd plural idealiniz idealleriniz
      3rd plural idealleri idealleri
      definite accusative
      singular plural
      1st singular idealimi ideallerimi
      2nd singular idealini ideallerini
      3rd singular idealini ideallerini
      1st plural idealimizi ideallerimizi
      2nd plural idealinizi ideallerinizi
      3rd plural ideallerini ideallerini
      dative
      singular plural
      1st singular idealime ideallerime
      2nd singular idealine ideallerine
      3rd singular idealine ideallerine
      1st plural idealimize ideallerimize
      2nd plural idealinize ideallerinize
      3rd plural ideallerine ideallerine
      locative
      singular plural
      1st singular idealimde ideallerimde
      2nd singular idealinde ideallerinde
      3rd singular idealinde ideallerinde
      1st plural idealimizde ideallerimizde
      2nd plural idealinizde ideallerinizde
      3rd plural ideallerinde ideallerinde
      ablative
      singular plural
      1st singular idealimden ideallerimden
      2nd singular idealinden ideallerinden
      3rd singular idealinden ideallerinden
      1st plural idealimizden ideallerimizden
      2nd plural idealinizden ideallerinizden
      3rd plural ideallerinden ideallerinden
      genitive
      singular plural
      1st singular idealimin ideallerimin
      2nd singular idealinin ideallerinin
      3rd singular idealinin ideallerinin
      1st plural idealimizin ideallerimizin
      2nd plural idealinizin ideallerinizin
      3rd plural ideallerinin ideallerinin
      Predicative forms
      singular plural
      1st singular idealim ideallerim
      2nd singular idealsin ideallersin
      3rd singular ideal
      idealdir       		idealler
      ideallerdir
      1st plural idealiz idealleriz
      2nd plural idealsiniz ideallersiniz
      3rd plural idealler ideallerdir

      Synonyms

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      Welsh

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      Alternative forms

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      Etymology

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      Borrowed from English ideal.[1]

      Pronunciation

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      Adjective

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      ideal (feminine singular ideal, plural ideal, not comparable)

      1. ideal[1][2]
        Synonym: delfrydol

      Noun

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      ideal m (plural idealau)

      1. ideal[1][2]
        Synonym: delfryd

      Mutation

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      Mutated forms of ideal
      radical soft nasal h-prothesis
      ideal unchanged unchanged hideal

      Note: Certain mutated forms of some words can never occur in standard Welsh.
      All possible mutated forms are displayed for convenience.

      References

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      1. ↑ 1.0 1.1 1.2 R. J. Thomas, G. A. Bevan, P. J. Donovan, A. Hawke, et al., editors (1950–present), “ideal”, in Geiriadur Prifysgol Cymru Online (in Welsh), University of Wales Centre for Advanced Welsh & Celtic Studies
      2. ↑ 2.0 2.1 Delyth Prys; J.P.M. Jones; Owain Davies; Gruffudd Prys (2006), Y Termiadur: termau wedi'u safoni; standardised terminology‎[2] (in Welsh), Cardiff: Awdurdod cymwysterau, cwricwlwm ac asesu Cymru (Qualifications curriculum & assessment authority for Wales), →ISBN
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