# supremum

## English

### Etymology

Borrowed from Latin supremum.

### Noun

supremum (plural supremums or suprema)

1. English Wikipedia has an article on:
Wikipedia
(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 ${\displaystyle S=\{t:t\in P:\forall x\in X,x\leq t\}}$ 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 ${\displaystyle \bigvee X}$.

## Czech

### Noun

supremum n

1. supremum
Antonym: infimum

#### Declension

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

## Finnish

### Pronunciation

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

### Noun

supremum

1. 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
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
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
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
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
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
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

## Latin

suprēmum

1. inflection of suprēmus:

### 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. supremum

#### Declension

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