supremumとは 意味・読み方・使い方

名詞

supremum (複数形 supremums または suprema)

1. (set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.
   • 2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 8:

     A sublattice of a lattice is a subset that is closed under pairwise infima and suprema.

   • 2010, James S. Howland, Basic Real Analysis, Jones & Bartlett Publishers, page 9:

     The best way to describe the supremum of S is to say that it wants to be the greatest element of S. In fact, if S has a greatest element, then that element is the supremum.

   • 2011, Andreas Löhne, Vector Optimization with Infimum and Supremum, Springer, page vii:

     The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.

使用する際の注意点

• Commonly denoted sup(X).
• The supremum of X may not exist, and, if it does, may not be an element of X.
• (order theory):
  • Formally: Let ${\displaystyle S=\{t:t\in P:\forall x\in X,x\leq t\}}$ be the set of upper bounds of X. Then sup(X), if it exists, is the element ${\displaystyle s\in S:\forall y\in S,s\leq y}$.
  • The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted ${\displaystyle \sup\{x,y\}}$ can also be written ${\displaystyle x\lor y}$. The supremum of a set may be denoted ${\displaystyle \sup(X)}$ or ${\displaystyle \bigvee X}$.

同意語

• (element of a set greater than または equal to all members of a given subset): least upper bound, LUB, sup

派生語

等位語

• infimum

参考

• maximum

Further reading

• Infimum and supremum on Wikipedia.
• Join and meet on Wikipedia.
• Lattice (order) on Wikipedia.
• Least-upper-bound property on Wikipedia.
• Upper and lower bounds on Wikipedia.

アナグラム

• supermum

Latin

形容詞

suprēmum

1. inflection of suprēmus:
   1. nominative/accusative/vocative neuter singular
   2. accusative masculine singular

参照

• “supremum”, in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
• supremum in Gaffiot, Félix (1934) Dictionnaire illustré latin-français, Hachette
• Carl Meißner; Henry William Auden (1894) Latin Phrase-Book‎^[1], London: Macmillan and Co.
  • (ambiguous) to depart this life: mortem (diem supremum) obire
    
  • (ambiguous) on one's last day: supremo vitae die
    
  • (ambiguous) to perform the last rites for a person: supremo officio in aliquem fungi
    
  • (ambiguous) to perform the last offices of affection: supremis officiis aliquem prosequi (vid sect. VI. 11., note Prosequi...)
    
  • (ambiguous) the last wishes of a deceased person: alicuius mortui voluntas (suprema)