Note: these 'words' (valid or invalid) are all the permutations of the word ideal. These words are obtained by scrambling the letters in ideal.
Definitions and meaning of ideal
ideal
Etymology
From Frenchidéal, from Late Latinideālis(“existing in idea”), from Latinidea(“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.
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;[…].
Teaching or relating to the doctrine of idealism.
(mathematics) Not actually present, but considered as present when limits at infinity are included.
Synonyms
(optimal):best, ideal, optimal, optimum
(flawless): see also Thesaurus:flawless
(of ideas):conceptual, notional
(existing only in mind):conceptual, imaginary
Derived terms
Related terms
Translations
Noun
ideal (pluralideals)
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., or a standard of excellence to aim at.
Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz
(algebra, ring theory) A subring closed under multiplication by its containing 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 .
(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).
1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,
An idealA 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.
(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 .
(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.
(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.
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
(order theory):order ideal
(type of Lie subalgebra):Lie ideal
Antonyms
(antonym(s) of "order theory"):filter
Hyponyms
(mathematics): maximal ideal, principal ideal
Derived terms
Translations
References
Further reading
Ideal (ring theory) on Wikipedia.Wikipedia
Ideal (order theory) on Wikipedia.Wikipedia
Ideal (set theory) on Wikipedia.Wikipedia
Ideal point on Wikipedia.Wikipedia
Ideal triangle on Wikipedia.Wikipedia
Lie algebra on Wikipedia.Wikipedia
Anagrams
Delia, Elida, ailed, ladie
Asturian
Etymology
From Latinideālis.
Pronunciation
IPA(key): /ideˈal/, [i.ð̞eˈal]
Adjective
ideal (epicene, pluralideales)
ideal
Noun
idealm (pluralideales)
ideal
Catalan
Etymology
Borrowed from Latinideālis.
Pronunciation
IPA(key): (Central, Balearic, Valencian)[i.ðeˈal]
Adjective
idealm or f (masculine and feminine pluralideals)
ideal
Derived terms
idealisme
idealista
idealitzar
idealment
Noun
idealm (pluralideals)
ideal
Galician
Etymology
From Latinideālis.
Pronunciation
Adjective
idealm or f (pluralideais)
ideal
Derived terms
idealmente
Noun
idealm (pluralideais)
ideal
German
Etymology
Borrowed from Late Latinideālis(“existing in idea”), from Latinidea(“idea”). Doublet of ideell.
(mathematics)ideal: a subring closed under multiplication by its containing ring.
Alternative forms
idiil
Affixed terms
Related terms
Further reading
“ideal” in Kamus Besar Bahasa Indonesia, Jakarta: Agency for Language Development and Cultivation — Ministry of Education, Culture, Research, and Technology of the Republic Indonesia, 2016.