# supremum
## English
### Etymology

Borrowed from Latin supremum.

### Pronunciation

* Audio (US):(file)

### Noun

supremum (plural supremums or suprema)
* English Wikipedia has an article on:supremumWikipedia (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
### 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: [ˈsuprɛːmum]

### Noun

supremum n
* (mathematics) supremum
    Antonym: infimum

#### Declension
### Further reading

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

## Finnish
### Etymology

Learned borrowing from Latin suprēmum.

### Pronunciation

* IPA: /ˈsupre(ː)mum/, [ˈs̠upre̞(ː)mum]
* Rhymes: -upremum
* Hyphenation: sup‧re‧mum

### Noun

supremum
* (mathematics) supremum

#### Declension

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
* inflection of suprēmus:
** nominative/accusative/vocative neuter singular
** 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)

## Swedish
### Noun

supremum n
* (mathematics) supremum

#### Declension

| | | nominative | genitive |
| singular | indefinite | supremum | supremums |
| singular | definite | supremumet | supremumets |
| plural | indefinite | supremum | supremums |
| plural | definite | supremumen | supremumens |