In mathematics, given a subset S of a partially ordered set T, the supremum (sup) of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound (lub or LUB). If the supremum exists, it may or may not belong to S. If the supremum exists, it is unique.

Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structure for which it is immediately clear what it means for an element to be "greater-than-or-equal-to" another element. The definition generalizes easily to the more abstract setting of order theory, where one considers arbitrary partially ordered sets.

The concept of supremum is not the same as the concepts of minimal upper bound, maximal element, or greatest element.

Supremum of a set of real numbers

In analysis, the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. An important property of the real numbers is its completeness: every nonempty subset of the set of real numbers that is bounded above has a supremum that is also a real number.

Examples

:$\backslash sup\; \backslash ,\; \backslash \{\; 1,\; 2,\; 3\; \backslash \}\; =\; 3\backslash ,$

:

:$\backslash sup\; \backslash ,\; \backslash \{\; (-1)^n\; -\; \backslash frac\{1\}\{n\}\; :\; n\; \backslash in\; \backslash mathbb\{N\}^\{*\}\; \backslash \}\; =\; 1\backslash ,$

:$\backslash sup\; \backslash ,\; \backslash \{\; a\; +\; b\; :\; a\; \backslash in\; A\; \backslash ,\; \backslash mbox\{and\}\; \backslash ,\; b\; \backslash in\; B\backslash \}\; =\; \backslash sup(A)\; +\; \backslash sup(B)\backslash ,$

:

In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.

One basic property of the supremum is

:$\backslash sup\; \backslash ,\; \backslash \{\; f(t)\; +\; g(t)\; :\; t\; \backslash in\; A\; \backslash \}\; \backslash le\; \backslash sup\; \backslash ,\; \backslash \{\; f(t)\; :\; t\; \backslash in\; A\; \backslash \}\; +\; \backslash sup\; \backslash ,\; \backslash \{\; g(t)\; :\; t\; \backslash in\; A\; \backslash \}$

for any functionals f and g.

If, in addition, we define sup(S) = −∞ when S is empty and sup(S) = +∞ when S is not bounded above, then every set of real numbers has a supremum under the affinely extended real number system.

:$\backslash sup\; \backslash mathbb\{Z\}\; =\; \backslash infty\backslash ,$

:$\backslash sup\; \backslash varnothing\; =\; -\backslash infty\backslash ,$

If the supremum belongs to the set, then it is the greatest element in the set. The term maximal element is synonymous as long as one deals with real numbers or any other totally ordered set.

To show that a = sup(S), one has to show that a is an upper bound for S and that any other upper bound for S is greater than a. Equivalently, one could alternatively show that a is an upper bound for S and that any number less than a is not an upper bound for S.

Suprema within partially ordered sets

Least upper bounds are important concepts in order theory, where they are also called joins (especially in lattice theory). As in the special case treated above, a supremum of a given set is just the least element of the set of its upper bounds, provided that such an element exists.

Formally, we have: For subsets S of arbitrary partially ordered sets (P, ≤), a supremum or least upper bound of S is an element u in P such that # x ≤ u for all x in S, and # for any v in P such that x ≤ v for all x in S it holds that u ≤ v.

Thus the supremum does not exist if there is no upper bound, or if the set of upper bounds has two or more elements of which none is a least element of that set. It can easily be shown that, if S has a supremum, then the supremum is unique (as the least element of any partially ordered set, if it exists, is unique): if u₁ and u₂ are both suprema of S then it follows that u₁ ≤ u₂ and u₂ ≤ u₁, and since ≤ is antisymmetric, one finds that u₁ = u₂.

If the supremum exists it may or may not belong to S. If S contains a greatest element, then that element is the supremum; and if not, then the supremum does not belong to S.

The dual concept of supremum, the greatest lower bound, is called infimum and is also known as meet.

If the supremum of a set S exists, it can be denoted as sup(S) or, which is more common in order theory, by $\backslash vee$S. Likewise, infima are denoted by inf(S) or $\backslash wedge$S. In lattice theory it is common to use the infimum/meet and supremum/join as binary operators; in this case $a\; \backslash vee\; b\; =\; \backslash sup~\backslash \{a,\; b\backslash \}$ (and similarly for infima).

A complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

In the sections below the difference between suprema, maximal elements, and minimal upper bounds is stressed. As a consequence of the possible absence of suprema, classes of partially ordered sets for which certain types of subsets are guaranteed to have least upper bound become especially interesting. This leads to the consideration of so-called completeness properties and to numerous definitions of special partially ordered sets.

Comparison with other order theoretical notions

Greatest elements

The distinction between the supremum of a set and the greatest element of a set may not be immediately obvious. The difference is that the greatest element must be a member of the set, whereas the supremum need not. For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number x, there is another negative real number x/2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

In general, this situation occurs for all subsets that do not contain a greatest element. In contrast, if a set does contain a greatest element, then it also has a supremum given by the greatest element.

Maximal elements

For an example where there are no greatest but still some maximal elements, consider the set of all subsets of the set of natural numbers (the powerset). We take the usual subset inclusion as an ordering, i.e. a set is greater than another set if it contains all elements of the other set. Now consider the set S of all sets that contain at most ten natural numbers. The set S has many maximal elements, i.e. elements for which there is no greater element. In fact, all sets with ten elements are maximal. However, the supremum of S is the (only and therefore least) set which contains all natural numbers. One can compute the least upper bound of a subset A of a powerset (i.e. A is a set of sets) by just taking the union of the elements of A.

Minimal upper bounds

Finally, a set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a totally ordered set, like the real numbers mentioned above, the concepts are the same.

As an example, let S be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from S together with the set of integers Z and the set of positive real numbers R+, ordered by subset inclusion as above. Then clearly both Z and R+ are greater than all finite sets of natural numbers. Yet, neither is R+ smaller than Z nor is the converse true: both sets are minimal upper bounds but none is a supremum.

Least-upper-bound property

The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness.

If an ordered set S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property. As noted above, the set R of all real numbers has the least-upper-bound property. Similarly, the set Z of integers has the least-upper-bound property; if S is a nonempty subset of Z and there is some number n such that every element s of S is less than or equal to n, then there is a least upper bound u for S, an integer that is an upper bound for S and is less than or equal to every other upper bound for S. A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set that lacks the least-upper-bound property is Q, the set of rational numbers. Let S be the set of all rational numbers q such that q² < 2. Then S has an upper bound (1000, for example, or 6) but no least upper bound in Q. For suppose p ∈ Q is an upper bound for S, so p² > 2. Then q = (2p+2)/(p + 2) is also an upper bound for S, and q < p. (To see this, note that q = p − (p² − 2)/(p + 2), and that p² − 2 is positive.) Another example is the Hyperreals; there is no least upper bound of the set of positive infinitesimals.

There is a corresponding 'greatest-lower-bound property'; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered set P every bounded subset has a supremum, this applies also, for any set X, in the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all x in X. For example, it applies for real functions, and, since these can be considered special cases of functions, for real n-tuples and sequences of real numbers.

See also

Infimum

Essential suprema and infima

Uniform norm (supremum norm)

Limit superior and limit inferior (supremum limit)

Specker sequence

References

Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.

External links

supremum (PlanetMath)

Supremum function at Wolfram

Category:Order theory

Guillaume Dufay (Du Fay, Du Fayt) (August 5, 1397? – November 27, 1474) was a Franco-Flemish composer of the early Renaissance. As the central figure in the Burgundian School, he was the most famous and influential composer in Europe in the mid-15th century.

Life

Early life

From the evidence of his will, he was probably born in Beersel, in the vicinity of Brussels. Circumstantial evidence claims him to have been born the illegitimate child of an unknown priest and a woman named Marie Du Fayt, though this suggestion is as of yet unproven. Marie moved with her son to Cambrai early in his life, staying with a relative who was a canon of the cathedral there. Soon Dufay's musical gifts were noticed by the cathedral authorities, who evidently gave him a thorough training in music; he studied with Rogier de Hesdin during the summer of 1409, and he was listed as a choirboy in the cathedral from 1409 to 1412. During those years he studied with Nicolas Malin, and the authorities must have been impressed with the boy's gifts because they gave him his own copy of Villedieu’s Doctrinale in 1411, a highly unusual event for one so young. In June 1414, at the age of only 16, he had already been given a benefice as chaplain at St. Géry, immediately adjacent to Cambrai. Later that year he probably went to the Council of Konstanz, staying possibly until 1418, at which time he returned to Cambrai.

From Cambrai to Italy and Savoy

From November 1418 to 1420 he was a subdeacon at Cambrai Cathedral. In 1420 he left Cambrai again, this time going to Rimini, and possibly Pesaro, where he worked for the Malatesta family. Although no records survive of his employment there, several compositions of his can be dated to this period; they contain references that make a residence in Italy reasonably certain. It was there that he met the composers Hugo and Arnold de Lantins, who were among the musicians of the Malatesta household. In 1424 Dufay again returned to Cambrai, this time because of the illness and subsequent death of the relative with whom his mother was staying. By 1426, however, he had gone back to Italy, this time to Bologna, where he entered the service of Cardinal Louis Aleman, the papal legate. While in Bologna he became a deacon, and by 1428 he was a priest. Cardinal Aleman was driven from Bologna by the rival Canedoli family in 1428, and Dufay also left at this time, going to Rome. He became a member of the Papal Choir, serving first Pope Martin V, and then after the death of Pope Martin in 1431, Pope Eugene IV. In 1434 he was appointed maistre de chappelle in Savoy, where he served Duke Amédée VIII; evidently he left Rome because of a crisis in the finances of the papal choir, and to escape the turbulence and uncertainty during the struggle between the papacy and the Council of Basel. Yet in 1435 he was again in the service of the papal chapel, but this time it was in Florence — Pope Eugene having been driven from Rome in 1434 by the establishment of an insurrectionary republic there, sympathetic to the Council of Basel and the Conciliar movement. In 1436 Dufay composed the festive motet Nuper rosarum flores, one of his most famous compositions, which was sung at the dedication of Brunelleschi's dome of the cathedral in Florence, where Eugene lived in exile.

During this period Dufay also began his long association with the Este family in Ferrara, some of the most important musical patrons of the Renaissance, and with which he probably had become acquainted during the days of his association with the Malatesta family; Rimini and Ferrara are not only geographically close, but the two families were related by marriage, and Dufay composed at least one ballade for Niccolò III, Marquis of Ferrara. In 1437 Dufay visited the town. When Niccolò died in 1441, the next Marquis maintained the contact with Dufay, and not only continued financial support for the composer but copied and distributed some of his music.

Return to Cambrai

The struggle between the papacy and the Council of Basel continued through the 1430s, and evidently Dufay realised that his own position might be threatened by the spreading conflict, especially since Pope Eugene was deposed in 1439 by the Council and replaced by Duke Amédée of Savoy himself, as Pope (Antipope) Felix V. At this time Dufay returned to his homeland, arriving in Cambrai by December of that year. In order to be a canon at Cambrai, he needed a law degree, which he obtained in 1437; he may have studied at University of Turin in 1436. One of the first documents mentioning him in Cambrai is dated December 27, 1440, when he received a delivery of 36 lots of wine for the feast of St. John the Evangelist.

Dufay was to remain in Cambrai through the 1440s, and during this time he was also in the service of the Duke of Burgundy. While in Cambrai he collaborated with Nicolas Grenon on a complete revision of the liturgical musical collection of the cathedral, which included writing an extensive collection of polyphonic music for services. In addition to his musical work, he was active in the general administration of the cathedral. In 1444 his mother Marie died, and was buried in the cathedral; and in 1445 Dufay moved into the house of the previous canon, which was to remain his primary residence for the rest of his life.

Travels to Savoy and Italy

After the abdication of the last antipope (Felix V) in 1449, his own former employer Duke Amédée VIII of Savoy, the struggle between different factions within the Church began to heal, and Dufay once again left Cambrai for points south. He went to Turin in 1450, shortly before the death of Duke Amédée, but returned to Cambrai later that year; and in 1452 he went back to Savoy yet again. This time he did not return to Cambrai for six years, and during that time he attempted to find either a benefice or an employment which would allow him to stay in Italy. Numerous compositions, including one of the four Lamentationes that he composed on the Fall of Constantinople in 1453, his famous mass based on Se la face ay pale, as well as a letter to Lorenzo de' Medici, survive from this period: but as he was unable to find a satisfactory position for his retirement, he returned north in 1458. While in Savoy he served more-or-less officially as choirmaster for Louis, Duke of Savoy, but he was more likely in a ceremonial role, since the records of the chapel never mention him.

Final years in Cambrai

When he returned to Cambrai for his final years, he was appointed canon of the cathedral. He was now the most renowned composer in Europe. Once again he established close ties to the court of Burgundy, and continued to compose music for them; in addition he received many visitors, including Busnois, Ockeghem, Tinctoris, and Loyset Compère, all of whom were decisive in the development of the polyphonic style of the next generation. During this period he probably wrote his mass based on L'homme armé, as well as the chanson on the same song; the latter composition may have been inspired by Philip the Good's call for a new crusade against the Turks, who had recently captured Constantinople. He also wrote a Requiem mass around 1460, which is lost.

After an illness of several weeks, Dufay died on November 27, 1474. He had requested that his motet Ave regina celorum be sung for him as he died, with pleas for mercy interpolated between verses of the antiphon, but time was insufficient for this to be arranged. Dufay was buried in the chapel of St. Etienne in the cathedral of Cambrai; his portrait was carved onto his tombstone. After the destruction of the cathedral the tombstone was lost, but it was found in 1859 (it was being used to cover a well), and is now in the Palais des Beaux Arts museum in Lille.

Music and influence

Dufay was among the most influential composers of the 15th century, and his music was copied, distributed and sung everywhere that polyphony had taken root. Almost all composers of the succeeding generations absorbed some elements of his style. The wide distribution of his music is all the more impressive considering that he died several decades before the availability of music printing.

Dufay wrote in most of the common forms of the day, including masses, motets, Magnificats, hymns, simple chant settings in fauxbourdon, and antiphons within the area of sacred music, and rondeaux, ballades, virelais and a few other chanson types within the realm of secular music. None of his surviving music is specifically instrumental, although instruments were certainly used for some of his secular music, especially for the lower parts; all of his sacred music is vocal. Instruments may have been used to reinforce the voices in actual performance for almost any portion of his output. In his lifetime, Dufay wrote seven complete masses, 28 individual Mass movements, 15 settings of chant used in Mass Propers, three Magnificats, two Benedicamus Domino settings, 15 antiphon settings (6 are Marian antiphons), 27 hymns, 22 motets (13 are isorhythmic) and 87 chansons. Assigning works to Dufay based on alleged stylistic similarities has been a favorite pastime of musicologists for at least a hundred years, judging from the copious literature on the subject.

Masses

Motets

Chant settings and fauxbourdon

Many of Dufay's compositions were simple settings of chant, obviously designed for liturgical use, likely as substitutes for the unadorned chant, and can be seen as chant harmonizations. Often the harmonization used a technique of parallel writing known as fauxbourdon, as in the following example, a setting of the Marian antiphon Ave maris stella:

Dufay may have been the first composer to use the term fauxbourdon to describe this style, which was prominent in 15th century liturgical music, especially that of the Burgundian school.

Secular music

Most of Dufay's secular songs follow the formes fixes (rondeau, ballade, and virelai), which dominated secular European music of the 14th and 15th centuries. He also wrote a handful of Italian ballate, almost certainly while he was in Italy. As is the case with his motets, many of the songs were written for specific occasions, and many are datable, thus supplying useful biographical information.

Most of his songs are for three voices, using a texture dominated by the highest voice; the other two voices, unsupplied with text, were likely played by instruments. Occasionally Dufay used four voices, but in a number of these songs the fourth voice was supplied by a later, usually anonymous, composer. Typically he used the rondeau form when writing love songs. His latest secular songs show influence from Busnois and Ockeghem, and the rhythmic and melodic differentiation between the voices is less; as in the work of other composers of the mid-15th century, he was beginning to tend towards the smooth polyphony which was to become the predominant style fifty years later.

A typical ballade is Resvellies vous et faites chiere lye, which was written in 1423 for the marriage of Carlo Malatesta and Vittoria di Lorenzo Colonna (Carlo was a son of Malatesta dei Sonetti, Lord of Pesaro. Vittoria was the niece of Pope Martin V). The musical form is aabC for each stanza, with C being the refrain. The musical setting emphasizes passages in the text which specifically refer to the couple being married.

Writings on music theory

Two written works by Dufay have been documented, but neither has survived. A note in the margin in a manuscript held in the Biblioteca Nazionale Palatina in Parma refers to a Musica which he wrote; no copy of the work itself has been found. Nineteenth-century musicologist François-Joseph Fétis claimed to have seen a sixteenth-century copy of a Tractatus de musica mensurata et de proportionibus by Dufay, last seen in a bookshop in London in 1824. The contents of neither work are known.

Influence

Dufay was one of the last composers to make use of medieval techniques such as isorhythm, but one of the first to use the harmonies, phrasing and expressive melodies characteristic of the early Renaissance. His compositions within the larger genres (masses, motets and chansons) are mostly similar to each other; his renown is largely due to what was perceived as his perfect control of the forms in which he worked, as well as his gift for memorable and singable melody. During the 15th century he was universally regarded as the greatest composer of the time, and that belief has largely persisted to the present day.

The early music ensemble Dufay Collective is named for the composer.

Sound samples

Blue Heron Renaissance Choir">Ecclesie Militantis performed by Blue Heron Renaissance Choir

Blue Heron Renaissance Choir">Rite Majorem performed by Blue Heron Renaissance Choir

References

David Fallows, "Dufay", revised edition. London, J.M. Dent & Sons, Ltd.: 1987. ISBN 0-460-02493-0

Massimo Mila: "Guillaume Dufay", ed. Simone Monge, Torino, Einaudi Editore, 1997, ISBN 8-806-14672-6

Massimo Mila: "Guillaume Dufay", Torino, G. Giappichelli Editore, 1972–73, 2 voll.

Charles Hamm, "Guillaume Dufay", in The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. 20 vol. London, Macmillan Publishers Ltd., 1980. ISBN 1-56159-174-2

Mark Lindley and Graeme Boone, "Euphony in Dufay. Harmonic 3rds and 6ths with explicit sharps in the early songs", in the 2004 Jahrbuch des Staatlichen Instituts für Musikforschung, Berlin

Gustave Reese, Music in the Renaissance. New York, W.W. Norton & Co., 1954. ISBN 0-393-09530-4

Alejandro Planchart: "Guillaume Du Fay", Grove Music Online ed. L. Macy (Accessed July 17, 2005), (subscription access)

Harold Gleason and Warren Becker, Music in the Middle Ages and Renaissance (Music Literature Outlines Series I). Bloomington, Indiana. Frangipani Press, 1986. ISBN 0-89917-034-X

Guillaume Dufay, Opera omnia (collected works in six volumes), ed. Heinrich Besseler with revisions by David Fallows. Corpus mensurabilis musicae CMM 1, American Institute of Musicology, 1951-1995. Further info and sample pages

Craig Wright, Dufay at Cambrai: Discoveries and Revisions. Journal of the American Musicological Society, XXVIII (I975)

F. Alberto Gallo, tr. Karen Eales, Music of the Middle Ages (II). Cambridge: Cambridge University Press. 1977 (original Italian edition) and 1985 (English). ISBN 0-521-28483-X

E. Dartus, Un grand musicien cambrésien - Guillaume Du Fay. Préface de Norbert Dufourcq. Extrait du tome XCIV des Mémoires de la Société d'Émulation de Cambrai. Cambrai, 1974

Van den Borren (I), Guillaume Du Fay, son importance .... Mémoires de l'Académie royale de Belgique, t. II, fasc. 2, 1926

Van den Borren (II), Guillaume Du Fay, centre de rayonnement... Bulletin de l'Institut historique belge de Rome, fasc. XX. Bruxelles, Rome, 1939

J. Chailley, Histoire musicale du Moyen Age. Paris, P. U. F.,1950

S. Baldi, Introduction to Il Conto dell'esecuzione del testamento e l'Inventario dei beni di Guillaume Dufay, Miscellanea di Studi 6, a cura di Alberto Basso, Torino, Centro Studi Piemontesi - Istituto per i Beni Musicali in Piemonte, 2006, 47-134.

S. Baldi, Guillaume Du Fay a Pinerolo, Bollettino della Società Storica Pinerolese, 25 (2008), 15-31 [with abstract in English].

E. Gasparini, Tra musica e architettura. Il Nuper rosarum flores di Dufay e la brunelleschiana cupola di Santa Maria del Fiore, «Musica Realtà», 88 (2009).

Notes

External links

Hear The Hilliard Ensemble perform Dufay's Moribus et genere; Vergene bella and Lamentatio sanctae matris ecclesiae Constantinopolitanae in London, 25 September 2010

Ave Regina Caelorum, the score at the classicaland.com

Category:1397 births Category:1474 deaths Category:Renaissance composers Category:Burgundian school composers