basis vs ideal

basis

noun
  • A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element. 

  • A starting point, base or foundation for an argument or hypothesis. 

  • A regular frequency. 

  • A physical base or foundation. 

  • In a vector space, a linearly independent set of vectors spanning the whole vector space. 

  • Amount paid for an investment, including commissions and other expenses. 

  • An underlying condition or circumstance. 

  • The difference between the cash price a dealer pays to a farmer for his produce and an agreed reference price, which is usually the futures price at which the given crop is trading at a commodity exchange. 

ideal

noun
  • 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. 

  • 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. 

  • A subring closed under multiplication by its containing ring. 

  • A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins). 

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

  • 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. 

adj
  • Not actually present, but considered as present when limits at infinity are included. 

  • Optimal; being the best possibility. 

  • Existing only in the mind; conceptual, imaginary. 

  • Pertaining to ideas, or to a given idea. 

  • Perfect, flawless, having no defects. 

  • Teaching or relating to the doctrine of idealism. 

How often have the words basis and ideal occurred in a corpus of books? (source: Google Ngram Viewer )