supremum

English

Etymology

Borrowed from Latin supremum.

Noun

supremum (plural supremums or 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.

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 $S=\{t:t\in P:\forall x\in X,x\leq t\}$ be the set of upper bounds of X. Then sup(X), if it exists, is the element $s\in S:\forall y\in S,s\leq y$.
  • The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted $\sup\{x,y\}$ can also be written $x\lor y$. The supremum of a set may be denoted $\sup(X)$ or $\bigvee X$.

Synonyms

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

Coordinate terms

• infimum

Derived terms

• essential supremum

 

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
• Spanish: supremo (es) 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

Czech

Pronunciation

• IPA^(key): [ˈsuprɛːmum]

Noun

supremum n

1. (mathematics) supremum

   Antonym: infimum

Declension

Declension of supremum (hard neuter foreign)

	singular	plural
nominative	supremum	suprema
genitive	suprema	suprem
dative	supremu	supremům
accusative	supremum	suprema
vocative	supremum	suprema
locative	supremu	supremech
instrumental	supremem	supremy

Further reading

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

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
abessive		supremumitta	supremumeitta
instructive		—	supremumein
comitative		See the possessive forms below.

Possessive forms of supremum (Kotus type 5/risti, no gradation)

first-person singular possessor singular plural nominative supremumini supremumini accusative nom. supremumini supremumini gen. supremumini genitive supremumini supremumieni partitive supremumiani supremumejani inessive supremumissani supremumeissani elative supremumistani supremumeistani illative supremumiini supremumeihini adessive supremumillani supremumeillani ablative supremumiltani supremumeiltani allative supremumilleni supremumeilleni essive supremuminani supremumeinani translative supremumikseni supremumeikseni abessive supremumittani supremumeittani instructive — — comitative — supremumeineni second-person singular possessor singular plural nominative supremumisi supremumisi accusative nom. supremumisi supremumisi gen. supremumisi genitive supremumisi supremumiesi partitive supremumiasi supremumejasi inessive supremumissasi supremumeissasi elative supremumistasi supremumeistasi illative supremumiisi supremumeihisi adessive supremumillasi supremumeillasi ablative supremumiltasi supremumeiltasi allative supremumillesi supremumeillesi essive supremuminasi supremumeinasi translative supremumiksesi supremumeiksesi abessive supremumittasi supremumeittasi instructive — — comitative — supremumeinesi first-person plural possessor singular plural nominative supremumimme supremumimme accusative nom. supremumimme supremumimme gen. supremumimme genitive supremumimme supremumiemme partitive supremumiamme supremumejamme inessive supremumissamme supremumeissamme elative supremumistamme supremumeistamme illative supremumiimme supremumeihimme adessive supremumillamme supremumeillamme ablative supremumiltamme supremumeiltamme allative supremumillemme supremumeillemme essive supremuminamme supremumeinamme translative supremumiksemme supremumeiksemme abessive supremumittamme supremumeittamme instructive — — comitative — supremumeinemme second-person plural possessor singular plural nominative supremuminne supremuminne accusative nom. supremuminne supremuminne gen. supremuminne genitive supremuminne supremumienne partitive supremumianne supremumejanne inessive supremumissanne supremumeissanne elative supremumistanne supremumeistanne illative supremumiinne supremumeihinne adessive supremumillanne supremumeillanne ablative supremumiltanne supremumeiltanne allative supremumillenne supremumeillenne essive supremuminanne supremumeinanne translative supremumiksenne supremumeiksenne abessive supremumittanne supremumeittanne instructive — — comitative — supremumeinenne third-person possessor singular plural nominative supremuminsa supremuminsa accusative nom. supremuminsa supremuminsa gen. supremuminsa genitive supremuminsa supremumiensa partitive supremumiaan supremumiansa supremumejaan supremumejansa inessive supremumissaan supremumissansa supremumeissaan supremumeissansa elative supremumistaan supremumistansa supremumeistaan supremumeistansa illative supremumiinsa supremumeihinsa adessive supremumillaan supremumillansa supremumeillaan supremumeillansa ablative supremumiltaan supremumiltansa supremumeiltaan supremumeiltansa allative supremumilleen supremumillensa supremumeilleen supremumeillensa essive supremuminaan supremuminansa supremumeinaan supremumeinansa translative supremumikseen supremumiksensa supremumeikseen supremumeiksensa abessive supremumittaan supremumittansa supremumeittaan supremumeittansa instructive — — comitative — supremumeineen supremumeinensa

Synonyms

• pienin yläraja

Antonyms

• infimum

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^[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)
    

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