supremum

supremum

English

Etymology

Borrowed from Latin supremum.

Noun

supremum (plural 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, Springer, 3rd Edition, 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.

Usage notes

• 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 ${\\displaystyle \\bigvee X}$.

Synonyms

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

Coordinate terms

• infimum

Translations

element of a set greater than or equal to all members of a given subset

Czech: supremum n Finnish: pienin yläraja, supremum (fi) French: supremum (fr) m German: Supremum (de) n Hebrew: סופרמום‎ m, חסם עליון‎ m Japanese: 上限 (ja) (じょうげん, jōgen) Polish: supremum (pl) n, kres górny (pl) m Swedish: supremum (sv)

See also

• maximum

Further reading

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

Anagrams

• supermum

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Czech

Pronunciation

• IPA^(key): [ˈsuprɛmum]

Noun

supremum n

1. (mathematics) supremum

Antonyms

• infimum

Further reading

• supremum in Akademický slovník cizích slov, 1995, at prirucka.ujc.cas.cz

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Finnish

Etymology

Learned borrowing from Latin suprēmum.

Pronunciation

• IPA^(key): /ˈsupre(ː)mum/, [ˈs̠upre̞(ː)mum]
• Rhymes: -upremum
• Syllabification^(key): sup‧re‧mum

Noun

supremum

1. (mathematics) supremum

Declension

Inflection of supremum (Kotus type 5/risti, no gradation)
nominative		supremum	supremumit
genitive		supremumin	supremumien
partitive		supremumia	supremumeja
illative		supremumiin	supremumeihin
		singular	plural
nominative		supremum	supremumit
accusative	nom.	supremum	supremumit
	gen.	supremumin
genitive		supremumin	supremumien
partitive		supremumia	supremumeja
inessive		supremumissa	supremumeissa
elative		supremumista	supremumeista
illative		supremumiin	supremumeihin
adessive		supremumilla	supremumeilla
ablative		supremumilta	supremumeilta
allative		supremumille	supremumeille
essive		supremumina	supremumeina
translative		supremumiksi	supremumeiksi
instructive		—	supremumein
abessive		supremumitta	supremumeitta
comitative		—	supremumeineen

Possessive forms of supremum (type risti)
possessor	singular	plural
1st person	supremumini	supremumimme
2nd person	supremumisi	supremuminne
3rd person	supremuminsa

Synonyms

• pienin yläraja

Antonyms

• infimum

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Latin

Adjective

suprēmum

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

References

• “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, 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)
    

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Swedish

Noun

supremum n

1. (mathematics) supremum

Declension

Declension of supremum
	Singular		Plural
	Indefinite	Definite	Indefinite	Definite
Nominative	supremum	supremumet	supremum	supremumen
Genitive	supremums	supremumets	supremums	supremumens