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Adalbert of Prague
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Early years
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Early years
Born as Vojtěch in 952 or in gord Libice, he belonged to the Slavnik clan, one of the two most powerful families in Bohemia. Events from his life were later recorded by a Bohemian priest Cosmas of Prague (1045–1125). Vojtěch's father was Slavník (d. 978–981), a duke ruling a province centred at Libice. His mother was Střezislava (d. 985–987), and according to David Kalhous belonged to the Přemyslid dynasty. He had five brothers: Soběslav, Spytimír, Dobroslav, Pořej, and Čáslav. Cosmas also refers to Radim (later Gaudentius) as a brother; who is believed to have been a half-brother by his father's liaison with another woman. After he survived a grave illness in childhood, his parents decided to dedicate him to the service of God. Adalbert was well educated, having studied for approximately ten years (970–80) in Magdeburg under Adalbert of Magdeburg. The young Vojtěch took his tutor's name "Adalbert" at his Confirmation.
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Adalbert of Prague
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Episcopacy
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Episcopacy
thumb|left|upright|210px|Monument to Adalbert and his brother Gaudentius, Libice nad Cidlinou, Czech Republic
thumb|left|upright|210px|Adalbert on a seal of the chapter of Gniezno Cathedral (Gnesen)
In 981 Adalbert of Magdeburg died, and his young protege Adalbert returned to Bohemia. Later Bishop Dietmar of Prague ordained him a Catholic priest. In 982, Bishop Dietmar died, and Adalbert, despite being under canonical age, was chosen to succeed him as Bishop of Prague. Amiable and somewhat worldly, he was not expected to trouble the secular powers by making excessive claims for the Church. Although Adalbert was from a wealthy family, he avoided comfort and luxury, and was noted for his charity and austerity. After six years of preaching and prayer, he had made little headway in evangelising the Bohemians, who maintained deeply embedded pagan beliefs.
Adalbert opposed the participation of Christians in the slave trade and complained of polygamy and idolatry, which were common among the people. Once he started to propose reforms he was met with opposition from both the secular powers and the clergy. His family refused to support Duke Boleslaus in an unsuccessful war against Poland. Adalbert was no longer welcome and eventually forced into exile. In 988 he went to Rome. He lived as a hermit at the Benedictine monastery of Saint Alexis. Five years later, Boleslaus requested that the Pope send Adalbert back to Prague, in hopes of securing his family's support. Pope John XV agreed, with the understanding that Adalbert was free to leave Prague if he continued to encounter entrenched resistance. Adalbert returned as bishop of Prague, where he was initially received with demonstrations of apparent joy. Together with a group of Italian Benedictine monks which brought with him, he founded in 14 January 993 a monastery in Břevnov (then situated westward from Prague, now part of the city), the second oldest monastery on Czech territory.
In 995, the Slavniks' former rivalry with the Přemyslids, who were allied with the powerful Bohemian clan of the Vršovids, resulted in the storming of the Slavnik town of Libice nad Cidlinou, which was led by the Přemyslid Boleslaus II the Pious. During the struggle four or five of Adalbert's brothers were killed. The Zlič Principality became part of the Přemyslids' estate. Adalbert unsuccessfully attempted to protect a noblewoman caught in adultery. She had fled to a convent, where she was killed. In upholding the right of sanctuary, Bishop Adalbert responded by excommunicating the murderers. Butler suggests that the incident was orchestrated by enemies of his family.
After this, Adalbert could not safely stay in Bohemia and escaped from Prague. Strachkvas was eventually appointed to be his successor. However, Strachkvas suddenly died during the liturgy at which he was to accede to his episcopal office in Prague. The cause of his death is still ambiguous. The Pope directed Adalbert to resume his see, but believing that he would not be allowed back, Adalbert requested a brief as an itinerant missionary.
Adalbert then traveled to Hungary and probably baptized Géza of Hungary and his son Stephen in Esztergom. Then he went to Poland where he was cordially welcomed by then-Duke Boleslaus I and installed as Bishop of Gniezno.
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Adalbert of Prague
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Mission and martyrdom in Prussia
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Mission and martyrdom in Prussia
thumb|Poland, Bohemia and Prussia during the reign of Boleslaus I
thumb|The execution of Saint Adalbert by the pagan Prussians, Gniezno Doors
Adalbert again relinquished his diocese, namely that of Gniezno, and set out as a missionary to preach to the inhabitants near Prussia. Bolesław I, Duke (and, later, King) of Poland, sent soldiers with Adalbert on his mission to the Prussians. The Bishop and his companions, entered Prussian territory and traveled along the coast of the Baltic Sea to Gdańsk. At the borders of the Polish realm, at the mouth of the Vistula River, his half-brother Radim (Gaudentius), Benedict-Bogusza (who was probably a Pole), and at least one interpreter, ventured out into Prussia alone, as Bolesław had only sent his soldiers to escort them to the border.
Adalbert achieved some success upon his arrival, however his arrival mostly caused strain upon the local Prussian populations. Partially this was because of the imperious manner with which he preached, but potentially because he preached utilizing a book. The Prussians had an oral society where communication was face to face. To the locals Adalbert reading from a book may have come off as a manifestation of an evil action. He was forced to leave this first village after being struck in the back of the head by an oar by a local chieftain, causing the pages of his book to scatter upon the ground. He and his companions then fled across a river.
In the next place that Adalbert tried to preach, his message was met with the locals banging their sticks upon the ground, calling for the death of Adalbert and his companions. Retreating once again Adalbert and his companions went to a market place of Truso (near modern-day Elbląg). Here they were met with a similar response as at the previous place. On the 23 April 997, after mass, while Adalbert and his companions lay in the grass while eating a snack, they were set upon by a pagan mob. The mob was led by a man named Sicco, possibly a pagan priest, who delivered the first blow against Adalbert, before the others joined in. They removed Adalbert's head from his body after he was dead, and mounted on a pole while they returned home. This encounter may also have taken place in Tenkitten and Fischhausen (now Primorsk, Kaliningrad Oblast, Russia). It is recorded that his body was bought back for its weight in gold by King Boleslaus I of Poland.
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Adalbert of Prague
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Veneration and relics
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Veneration and relics
thumb|upright|Silver coffin of Adalbert, Cathedral in Gniezno
thumb|upright|Canonical cross of Saint Adalbert by Giennadij Jerszow. Collegiate Capitol in Gdańsk. Silver-Gold 2011
thumb|upright|Statue of Saint Adalbert in Prague
A few years after his martyrdom, Adalbert was canonized as Saint Adalbert of Prague. His life was written in Vita Sancti Adalberti Pragensis by various authors, the earliest being traced to imperial Aachen and the Bishop of Liège, Notger von Lüttich, although it was previously assumed that the Roman monk John Canaparius wrote the first Vita in 999. Another famous biographer of Adalbert was Bruno of Querfurt who wrote a hagiography of him in 1001–4.
Notably, the Přemyslid rulers of Bohemia initially refused to ransom Adalbert's body from the Prussians who murdered him, and therefore it was purchased by Poles. This fact may be explained by Adalbert's belonging to the Slavniks family which was rival to the Přemyslids. Thus Adalbert's bones were preserved in Gniezno, which assisted Boleslaus I of Poland in increasing Polish political and diplomatic power in Europe.
According to Bohemian accounts, in 1039 the Bohemian Duke Bretislav I looted the bones of Adalbert from Gniezno in a raid and translated them to Prague. According to Polish accounts, however, he stole the wrong relics, namely those of Gaudentius, while the Poles concealed Adalbert's relics which remain in Gniezno. In 1127 his severed head, which was not in the original purchase according to Roczniki Polskie, was discovered and translated to Gniezno. In 1928, one of the arms of Adalbert, which Bolesław I had given to Holy Roman Emperor Otto III in 1000, was added to the bones preserved in Gniezno. Therefore, today Adalbert has two elaborate shrines in the Prague Cathedral and Royal Cathedral of Gniezno, each of which claims to possess his relics, but which of these bones are his authentic relics is unknown. For example, pursuant to both claims two skulls are attributed to Adalbert. The one in Gniezno was stolen in 1923.
The massive bronze doors of Gniezno Cathedral, dating from around 1175, are decorated with eighteen reliefs of scenes from Adalbert's life. They are the only Romanesque ecclesiastical doors in Europe depicting a cycle illustrating the life of a saint, and therefore are a precious relic documenting Adalbert's martyrdom. We can read that door literally and theologically.
The one thousandth anniversary of Adalbert's martyrdom was on 23 April 1997. It was commemorated in Poland, the Czech Republic, Germany, Russia, and other nations. Representatives of Catholic, Eastern Orthodox, and Evangelical churches traveled on a pilgrimage to Adalbert's tomb located in Gniezno. Pope John Paul II visited the cathedral and celebrated a liturgy there in which heads of seven European nations and approximately one million faithful participated.
A ten-meter cross was erected near the village of Beregovoe (formerly Tenkitten), Kaliningrad Oblast, where Adalbert is thought to have been martyred by the Prussians.
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Adalbert of Prague
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Feast day
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Feast day
25 January – commemoration of translation of relics to Church of Saint Roch,
22 April – commemoration in Diocese of Innsbruck,
22 April – commemoration in Catholic Church in England and Wales,
23 April – commemoration of death anniversary,
14 May – commemoration of consecration of church in Aachen
25 August – commemoration of translation of relics from Gniezno to Prague (1039)
26 August – commemoration of translation of relics to Wrocław
20 October – commemoration of translation of relics to Gniezno (1090)
22 October – commemoration of translation of relics to Gniezno
6 November – commemoration of translation of relics to Esztergom,
He is also commemorated on 23 April by Evangelical Church in Germany and Eastern Orthodox Church.
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Adalbert of Prague
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In popular culture and society
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In popular culture and society
The Dagmar and Václav Havel VIZE 97 Foundation Prize, given annually to a distinguished thinker "whose work exceeds the traditional framework of scientific knowledge, contributes to the understanding of science as an integral part of general culture and is concerned with unconventional ways of asking fundamental questions about cognition, being and human existence"
includes a massive replica of Adalbert's crozier by Czech artist Jiří Plieštík.
St. Vojtech Fellowship was established in 1870 by Slovak Catholic priest Andrej Radlinský. It had facilitated Slovak Catholic thinkers and authors, continuing to publish religious original works and translations to this day. It is the official publishing body of Episcopal Conference of Slovakia.
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Adalbert of Prague
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Churches and parishes named for Adalbert
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Churches and parishes named for Adalbert
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Adalbert of Prague
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See also
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See also
History of the Czech lands in the Middle Ages
History of Poland (966–1385)
Congress of Gniezno
Gniezno Doors
Adalbert of Magdeburg
Saint Adalbert of Prague, patron saint archive
Statue of Adalbert of Prague, Charles Bridge
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Adalbert of Prague
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References
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References
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Adalbert of Prague
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Sources
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Sources
Donald Attwater and Catherine R. John, The Penguin Dictionary of Saints, Third Edition (New York: Penguin Books, 1993); .
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Adalbert of Prague
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External links
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External links
Category:950s births
Category:997 deaths
Category:10th-century bishops in Bohemia
Category:Nobility from medieval Bohemia
Category:Slavník dynasty
Category:People from Nymburk District
Category:Czech Christian missionaries
Category:Czech Roman Catholic saints
Category:Burials at St. Vitus Cathedral
Category:Burials at Gniezno Cathedral
Category:Saints from medieval Bohemia
Category:Polish Roman Catholic saints
Category:10th-century Christian saints
Category:10th-century Christian martyrs
Category:10th century in Hungary
Category:10th century in Poland
Category:Christian missionaries in Europe
Category:Patron saints of Poland
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Adalbert of Prague
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Table of Content
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Short description, Life, Early years, Episcopacy, Mission and martyrdom in Prussia, Veneration and relics, Feast day, In popular culture and society, Churches and parishes named for Adalbert, See also, References, Sources, External links
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Ælfheah of Canterbury
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Short description
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Ælfheah ( – 19 April 1012), more commonly known today as Alphege, was an Anglo-Saxon Bishop of Winchester, later Archbishop of Canterbury. He became an anchorite before being elected abbot of Bath Abbey. His reputation for piety and sanctity led to his promotion to the episcopate and, eventually, to his becoming archbishop. Ælfheah furthered the cult of Dunstan and also encouraged learning. He was captured by Viking raiders in 1011 during the siege of Canterbury and killed by them the following year after refusing to allow himself to be ransomed. Ælfheah was canonised as a saint in 1078. Thomas Becket, a later Archbishop of Canterbury, prayed to Ælfheah just before his murder in Canterbury Cathedral in 1170.
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Ælfheah of Canterbury
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Life
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Life
Ælfheah was born around 953,Rumble "From Winchester to Canterbury" Leaders of the Anglo-Saxon Church p. 165 supposedly in Weston on the outskirts of Bath, Accessed 14 August 2009 and became a monk early in life. He first entered the monastery of Deerhurst, but then moved to Bath, where he became an anchorite. He was noted for his piety and austerity and rose to become abbot of Bath Abbey.Knowles, et al. Heads of Religious Houses, England and Wales pp. 28, 241 The 12th-century chronicler, William of Malmesbury recorded that Ælfheah was a monk and prior at Glastonbury Abbey,Rumble "From Winchester to Canterbury" Leaders of the Anglo-Saxon Church p. 166 but this is not accepted by all historians. Indications are that Ælfheah became abbot at Bath by 982, perhaps as early as around 977. He perhaps shared authority with his predecessor Æscwig after 968.
Probably due to the influence of Dunstan, the Archbishop of Canterbury (959–988), Ælfheah was elected Bishop of Winchester in 984,Fryde, et al. Handbook of British Chronology p. 223Barlow English Church 1000–1066 p. 109 footnote 5 and was consecrated on 19 October that year. While bishop, he was largely responsible for the construction of a large organ in the cathedral, audible from over a mile (1600 m) away and said to require more than 24 men to operate. He also built and enlarged the city's churches,Hindley A Brief History of the Anglo-Saxons pp. 304–305 and promoted the cult of Swithun and his predecessor, Æthelwold of Winchester. One act promoting Æthelwold's cult was the translation of Æthelwold's body to a new tomb in the cathedral at Winchester, which Ælfheah presided over on 10 September 996.Rumble "From Winchester to Canterbury" Leaders of the Anglo-Saxon Church p. 167
Following a Viking raid in 994, a peace treaty was agreed with one of the raiders, Olaf Tryggvason. Besides receiving danegeld, Olaf converted to ChristianityStenton Anglo-Saxon England p. 378 and undertook never to raid or fight the English again.Williams Æthelred the Unready p. 47 Ælfheah may have played a part in the treaty negotiations, and it is certain that he confirmed Olaf in his new faith.Leyser "Ælfheah" Oxford Dictionary of National Biography
In 1006, Ælfheah succeeded Ælfric as Archbishop of Canterbury,Walsh New Dictionary of Saints p. 28Fryde, et al. Handbook of British Chronology p. 214 taking Swithun's head with him as a relic for the new location. He went to Rome in 1007 to receive his pallium—symbol of his status as an archbishop—from Pope John XVIII, but was robbed during his journey.Barlow English Church 1000–1066 pp. 298–299 footnote 7 While at Canterbury, he promoted the cult of Dunstan, ordering the writing of the second Life of Dunstan, which Adelard of Ghent composed between 1006 and 1011.Barlow English Church 1000–1066 p. 62 He also introduced new practices into the liturgy, and was instrumental in the Witenagemot's recognition of Wulfsige of Sherborne as a saint in about 1012.Barlow English Church 1000–1066 p. 223
Ælfheah sent Ælfric of Eynsham to Cerne Abbey to take charge of its monastic school.Stenton Anglo-Saxon England p. 458 He was present at the council of May 1008 at which Wulfstan II, Archbishop of York, preached his Sermo Lupi ad Anglos (The Sermon of the Wolf to the English), castigating the English for their moral failings and blaming the latter for the tribulations afflicting the country.Fletcher Bloodfeud p. 94
In 1011, the Danes again raided England, and from 8–29 September they laid siege to Canterbury. Aided by the treachery of Ælfmaer, whose life Ælfheah had once saved, the raiders succeeded in sacking the city.Williams Æthelred the Unready pp. 106–107 Ælfheah was taken prisoner and held captive for seven months.Hindley Brief History of the Anglo-Saxons p. 301 Godwine (Bishop of Rochester), Leofrun (abbess of St Mildrith's), and the king's reeve, Ælfweard were captured also, but the abbot of St Augustine's Abbey, Ælfmær, managed to escape. Canterbury Cathedral was plundered and burned by the Danes following Ælfheah's capture.Barlow English Church 1000–1066 pp. 209–210
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Ælfheah of Canterbury
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Death
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Death
thumb|right|Memorial to Saint Ælfheah inside the Church of Saint Alfege, Greenwich
Ælfheah refused to allow a ransom to be paid for his freedom, and as a result was killed on 19 April 1012 at Greenwich, reputedly on the site of St Alfege's Church. The account of Ælfheah's death appears in the E version of the Anglo-Saxon Chronicle:
Ælfheah was the first Archbishop of Canterbury to die a violent death.Fletcher Bloodfeud p. 78 A contemporary report tells that Thorkell the Tall attempted to save Ælfheah from the mob about to kill him by offering everything he owned except for his ship, in exchange for Ælfheah's life; Thorkell's presence is not mentioned in the Anglo-Saxon Chronicle, however.Williams Æthelred the Unready pp. 109–110 Some sources record that the final blow, with the back of an axe, was delivered as an act of kindness by a Christian convert known as "Thrum". Ælfheah was buried in Old St Paul's Cathedral. In 1023, his body was moved by King Cnut to Canterbury, with great ceremony.Hindley Brief History of the Anglo-Saxons pp. 309–310 Thorkell the Tall was appalled at the brutality of his fellow raiders, and switched sides to the English king Æthelred the Unready following Ælfheah's death.Stenton Anglo-Saxon England p. 383
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Ælfheah of Canterbury
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Veneration
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Veneration
thumb|upright=.8|An 1868 statue on the West Front of Salisbury Cathedral by James Redfern, showing Ælfheah holding the stones used in his martyrdom.
Pope Gregory VII canonised Ælfheah in 1078, with a feast day of 19 April.Delaney Dictionary of Saints pp. 29–30 Lanfranc, the first post-Conquest archbishop, was dubious about some of the saints venerated at Canterbury. He was persuaded of Ælfheah's sanctity,Williams English and the Norman Conquest p. 137 but Ælfheah and Augustine of Canterbury were the only pre-conquest Anglo-Saxon archbishops kept on Canterbury's calendar of saints.Stenton Anglo-Saxon England p. 672 Ælfheah's shrine, which had become neglected, was rebuilt and expanded in the early 12th century under Anselm of Canterbury, who was instrumental in retaining Ælfheah's name in the church calendar.Brooke Popular Religion in the Middle Ages p. 40Southern "St Anselm and his English Pupils" Mediaeval and Renaissance Studies After the 1174 fire in Canterbury Cathedral, Ælfheah's remains, together with those of Dunstan were placed around the high altar, at which Thomas Becket is said to have commended his life into Ælfheah's care shortly before his martyrdom during the Becket controversy. The new shrine was sealed in lead,Nilson Cathedral Shrines p. 33 and was north of the high altar, sharing the honour with Dunstan's shrine, which was located south of the high altar.Nilson Cathedral Shrines pp. 66–67 A Life of Saint Ælfheah in prose and verse was written by a Canterbury monk named Osbern, at Lanfranc's request. The prose version has survived, but the Life is very much a hagiography; many of the stories it contains have obvious Biblical parallels, making them suspect as a historical record.
In the late medieval period, Ælfheah's feast day was celebrated in Scandinavia, perhaps because of the saint's connection with Cnut.Blair "Handlist of Anglo-Saxon Saints" Local Saints and Local Churches p. 504 Few church dedications to him are known, with most of them occurring in Kent and one each in London and Winchester; as well as St Alfege's Church in Greenwich, a nearby hospital (1931–1968) was named after him. In Kent, there are two 12th-century parish churches dedicated to St Alphege at Seasalter and Canterbury. Reputedly his body lay in these churches overnight on his way back to Canterbury Cathedral for burial.Histories in the parish collection at Canterbury Cathedral Archives and Library. In the town of Solihull in the West Midlands, St Alphege Church is dedicated to Ælfheah dating back to approximately 1277. In 1929, a new Roman Catholic church in Bath, the Church of Our Lady & St Alphege, was designed by Giles Gilbert Scott in homage to the ancient Roman church of Santa Maria in Cosmedin, and dedicated to Ælfheah under the name of Alphege. Accessed 30 August 2009 St George the Martyr with St Alphege & St Jude stands in the Borough in London.
Artistic representations of Ælfheah often depict him holding a pile of stones in his chasuble, a reference to his martyrdom.Audsley Handbook of Christian Symbolism p. 125
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Ælfheah of Canterbury
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Notes
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Notes
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Ælfheah of Canterbury
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Citations
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Citations
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Ælfheah of Canterbury
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References
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References
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Ælfheah of Canterbury
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Further reading
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Further reading
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Ælfheah of Canterbury
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External links
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External links
Category:950s births
Category:1012 deaths
Category:Clergy from Bath, Somerset
Category:Anglo-Saxon saints
Category:Archbishops of Canterbury
Category:Bishops of Winchester
Category:Martyred Roman Catholic priests
Category:11th-century Christian saints
Category:11th-century Christian martyrs
Category:Incorrupt saints
Category:Year of birth uncertain
Category:11th-century English Roman Catholic archbishops
Category:Anglican saints
Category:Canonizations by Pope Gregory VII
Category:English saints
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Ælfheah of Canterbury
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Table of Content
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Short description, Life, Death, Veneration, Notes, Citations, References, Further reading, External links
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Associative algebra
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Short description
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In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K). The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication.
A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring.
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Every ring is an associative algebra over its center and over the integers.
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Associative algebra
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Definition
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Definition
Let R be a commutative ring (so R could be a field). An associative R-algebra A (or more simply, an R-algebra A) is a ring A
that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies
for all r in R and x, y in the algebra. (This definition implies that the algebra, being a ring, is unital, since rings are supposed to have a multiplicative identity.)
Equivalently, an associative algebra A is a ring together with a ring homomorphism from R to the center of A. If f is such a homomorphism, the scalar multiplication is (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by . (See also below).
Every ring is an associative Z-algebra, where Z denotes the ring of the integers.
A is an associative algebra that is also a commutative ring.
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Associative algebra
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As a monoid object in the category of modules
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As a monoid object in the category of modules
The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map
.
The associativity then refers to the identity:
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Associative algebra
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From ring homomorphisms
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From ring homomorphisms
An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism whose image lies in the center of A, we can make A an R-algebra by defining
for all and . If A is an R-algebra, taking , the same formula in turn defines a ring homomorphism whose image lies in the center.
If a ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism .
The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms for a fixed R, i.e., commutative R-algebras, and whose morphisms are ring homomorphisms that are under R; i.e., is (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R.
How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring.
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Associative algebra
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Algebra homomorphisms
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Algebra homomorphisms
A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, is an associative algebra homomorphism if
The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.
The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings.
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Associative algebra
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Examples
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Examples
The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
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Associative algebra
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Algebra
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Algebra
Any ring A can be considered as a Z-algebra. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore, rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.
Any ring of characteristic n is a (Z/nZ)-algebra in the same way.
Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining .
Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated, free R-module.
In particular, the square n-by-n matrices with entries from the field K form an associative algebra over K.
The complex numbers form a 2-dimensional commutative algebra over the real numbers.
The quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
Every polynomial ring is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set .
The free R-algebra on a set E is an algebra of "polynomials" with coefficients in R and noncommuting indeterminates taken from the set E.
The tensor algebra of an R-module is naturally an associative R-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor that maps an R-module to its tensor algebra is left adjoint to the functor that sends an R-algebra to its underlying R-module (forgetting the multiplicative structure).
Given a module M over a commutative ring R, the direct sum of modules has a structure of an R-algebra by thinking M consists of infinitesimal elements; i.e., the multiplication is given as . The notion is sometimes called the algebra of dual numbers.
A quasi-free algebra, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.
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Associative algebra
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Representation theory
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Representation theory
The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
If G is a group and R is a commutative ring, the set of all functions from G to R with finite support form an R-algebra with the convolution as multiplication. It is called the group algebra of G. The construction is the starting point for the application to the study of (discrete) groups.
If G is an algebraic group (e.g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many structures of G translate to those of A.
A quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
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Associative algebra
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Analysis
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Analysis
Given any Banach space X, the continuous linear operators form an associative algebra (using composition of operators as multiplication); this is a Banach algebra.
Given any topological space X, the continuous real- or complex-valued functions on X form a real or complex associative algebra; here the functions are added and multiplied pointwise.
The set of semimartingales defined on the filtered probability space forms a ring under stochastic integration.
The Weyl algebra
An Azumaya algebra
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Associative algebra
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Geometry and combinatorics
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Geometry and combinatorics
The Clifford algebras, which are useful in geometry and physics.
Incidence algebras of locally finite partially ordered sets are associative algebras considered in combinatorics.
The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra.
A differential graded algebra is an associative algebra together with a grading and a differential. For example, the de Rham algebra , where consists of differential p-forms on a manifold M, is a differential graded algebra.
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Associative algebra
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Mathematical physics
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Mathematical physics
A Poisson algebra is a commutative associative algebra over a field together with a structure of a Lie algebra so that the Lie bracket satisfies the Leibniz rule; i.e., .
Given a Poisson algebra , consider the vector space of formal power series over . If has a structure of an associative algebra with multiplication such that, for ,
then is called a deformation quantization of .
A quantized enveloping algebra. The dual of such an algebra turns out to be an associative algebra (see ) and is, philosophically speaking, the (quantized) coordinate ring of a quantum group.
Gerstenhaber algebra
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Associative algebra
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Constructions
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Constructions
Subalgebras A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.
Quotient algebras Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since . This gives the quotient ring the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.
Direct products The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.
Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras.
Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring R and any ring A the tensor product R ⊗Z A can be given the structure of an R-algebra by defining . The functor which sends A to is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings.
Free algebra A free algebra is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.
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Associative algebra
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Dual of an associative algebra
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Dual of an associative algebra
Let A be an associative algebra over a commutative ring R. Since A is in particular a module, we can take the dual module A* of A. A priori, the dual A* need not have a structure of an associative algebra. However, A may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra.
For example, take A to be the ring of continuous functions on a compact group G. Then, not only A is an associative algebra, but it also comes with the co-multiplication and co-unit . The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual A* is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see below).
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Associative algebra
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Enveloping algebra
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Enveloping algebra
Given an associative algebra A over a commutative ring R, the enveloping algebra Ae of A is the algebra or , depending on authors.
Note that a bimodule over A is exactly a left module over Ae.
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Associative algebra
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Separable algebra
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Separable algebra
Let A be an algebra over a commutative ring R. Then the algebra A is a right module over with the action . Then, by definition, A is said to separable if the multiplication map splits as an Ae-linear map, where is an Ae-module by . Equivalently,
A is separable if it is a projective module over ; thus, the -projective dimension of A, sometimes called the bidimension of A, measures the failure of separability.
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Associative algebra
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Finite-dimensional algebra
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Finite-dimensional algebra
Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring.
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Associative algebra
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Commutative case
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Commutative case
As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent
is separable.
is reduced, where is some algebraic closure of k.
for some n.
is the number of -algebra homomorphisms .
Let , the profinite group of finite Galois extensions of k. Then is an anti-equivalence of the category of finite-dimensional separable k-algebras to the category of finite sets with continuous -actions.
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Associative algebra
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Noncommutative case
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Noncommutative case
Since a simple Artinian ring is a (full) matrix ring over a division ring, if A is a simple algebra, then A is a (full) matrix algebra over a division algebra D over k; i.e., . More generally, if A is a semisimple algebra, then it is a finite product of matrix algebras (over various division k-algebras), the fact known as the Artin–Wedderburn theorem.
The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
The Wedderburn principal theorem states: for a finite-dimensional algebra A with a nilpotent ideal I, if the projective dimension of as a module over the enveloping algebra is at most one, then the natural surjection splits; i.e., A contains a subalgebra B such that is an isomorphism. Taking I to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for Lie algebras.
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Associative algebra
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Lattices and orders
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Lattices and orders
Let R be a Noetherian integral domain with field of fractions K (for example, they can be Z, Q). A lattice L in a finite-dimensional K-vector space V is a finitely generated R-submodule of V that spans V; in other words, .
Let AK be a finite-dimensional K-algebra. An order in AK is an R-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., Z is a lattice in Q but not an order (since it is not an algebra).
A maximal order is an order that is maximal among all the orders.
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Associative algebra
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Related concepts
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Related concepts
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Associative algebra
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Coalgebras
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Coalgebras
An associative algebra over K is given by a K-vector space A endowed with a bilinear map having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism identifying the scalar multiples of the multiplicative identity. If the bilinear map is reinterpreted as a linear map (i.e., morphism in the category of K-vector spaces) (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form and one of the form ) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above.
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Associative algebra
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Representations
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Representations
A representation of an algebra A is an algebra homomorphism from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, for all x and y in A), and that ρ sends the unit of A to the unit of End(V) (that is, to the identity endomorphism of V).
If A and B are two algebras, and and are two representations, then there is a (canonical) representation of the tensor product algebra on the vector space . However, there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.
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Associative algebra
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Motivation for a Hopf algebra
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Motivation for a Hopf algebra
Consider, for example, two representations and . One might try to form a tensor product representation according to how it acts on the product vector space, so that
However, such a map would not be linear, since one would have
for . One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism , and defining the tensor product representation as
Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
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Associative algebra
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Motivation for a Lie algebra
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Motivation for a Lie algebra
One can try to be more clever in defining a tensor product. Consider, for example,
so that the action on the tensor product space is given by
.
This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
.
But, in general, this does not equal
.
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a Lie algebra.
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Associative algebra
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Non-unital algebras
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Non-unital algebras
Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.
One example of a non-unital associative algebra is given by the set of all functions whose limit as x nears infinity is zero.
Another example is the vector space of continuous periodic functions, together with the convolution product.
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Associative algebra
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See also
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See also
Abstract algebra
Algebraic structure
Algebra over a field
Sheaf of algebras, a sort of an algebra over a ringed space
Deligne's conjecture on Hochschild cohomology
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Associative algebra
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Notes
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Notes
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Associative algebra
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Citations
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Citations
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Associative algebra
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References
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References
James Byrnie Shaw (1907) A Synopsis of Linear Associative Algebra, link from Cornell University Historical Math Monographs.
Ross Street (1998) Quantum Groups: an entrée to modern algebra, an overview of index-free notation.
Category:Algebras
Category:Algebraic geometry
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Associative algebra
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Table of Content
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Short description, Definition, As a monoid object in the category of modules, From ring homomorphisms, Algebra homomorphisms, Examples, Algebra, Representation theory, Analysis, Geometry and combinatorics, Mathematical physics, Constructions, Dual of an associative algebra, Enveloping algebra, Separable algebra, Finite-dimensional algebra, Commutative case, Noncommutative case, Lattices and orders, Related concepts, Coalgebras, Representations, Motivation for a Hopf algebra, Motivation for a Lie algebra, Non-unital algebras, See also, Notes, Citations, References
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Axiom of regularity
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short description
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In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:
The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
The axiom was originally formulated by von Neumann; it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo. Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity. However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on
Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.
In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.
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Axiom of regularity
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Elementary implications of regularity
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Elementary implications of regularity
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Axiom of regularity
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No set is an element of itself
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No set is an element of itself
Let A be a set, and apply the axiom of regularity to {A}, which is a set by the axiom of pairing. We see that there must be an element of {A} which is disjoint from {A}. Since the only element of {A} is A, it must be that A is disjoint from {A}. So, since , we cannot have A an element of A (by the definition of disjoint).
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Axiom of regularity
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No infinite descending sequence of sets exists
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No infinite descending sequence of sets exists
Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for each n. Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema of replacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the definition of S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets. Since our supposition led to a contradiction, there must not be any such function, f.
The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.
Notice that this argument only applies to functions f that can be represented as sets as opposed to undefinable classes. The hereditarily finite sets, Vω, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom of infinity). So if one forms a non-trivial ultrapower of Vω, then it will also satisfy the axiom of regularity. The resulting model will contain elements, called non-standard natural numbers, that satisfy the definition of natural numbers in that model but are not really natural numbers. They are "fake" natural numbers which are "larger" than any actual natural number. This model will contain infinite descending sequences of elements. For example, suppose n is a non-standard natural number, then and , and so on. For any actual natural number k, . This is an unending descending sequence of elements. But this sequence is not definable in the model and thus not a set. So no contradiction to regularity can be proved.
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Axiom of regularity
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Simpler set-theoretic definition of the ordered pair
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Simpler set-theoretic definition of the ordered pair
The axiom of regularity enables defining the ordered pair (a,b) as {a,{a,b}}; see ordered pair for specifics. This definition eliminates one pair of braces from the canonical Kuratowski definition (a,b) = {{a},{a,b}}.
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Axiom of regularity
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Every set has an ordinal rank
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Every set has an ordinal rank
This was actually the original form of the axiom in von Neumann's axiomatization.
Suppose x is any set. Let t be the transitive closure of {x}. Let u be the subset of t consisting of unranked sets. If u is empty, then x is ranked and we are done. Otherwise, apply the axiom of regularity to u to get an element w of u which is disjoint from u. Since w is in u, w is unranked. w is a subset of t by the definition of transitive closure. Since w is disjoint from u, every element of w is ranked. Applying the axioms of replacement and union to combine the ranks of the elements of w, we get an ordinal rank for w, to wit . This contradicts the conclusion that w is unranked. So the assumption that u was non-empty must be false and x must have rank.
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Axiom of regularity
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For every two sets, only one can be an element of the other
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For every two sets, only one can be an element of the other
Let X and Y be sets. Then apply the axiom of regularity to the set {X,Y} (which exists by the axiom of pairing). We see there must be an element of {X,Y} which is also disjoint from it. It must be either X or Y. By the definition of disjoint then, we must have either Y is not an element of X or vice versa.
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Axiom of regularity
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The axiom of dependent choice and no infinite descending sequence of sets implies regularity
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The axiom of dependent choice and no infinite descending sequence of sets implies regularity
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by , which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
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Axiom of regularity
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Regularity and the rest of ZF(C) axioms
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Regularity and the rest of ZF(C) axioms
Regularity was shown to be relatively consistent with the rest of ZF by Skolem and von Neumann, meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent.For his proof in modern notation, see for instance.
The axiom of regularity was also shown to be independent from the other axioms of ZFC, assuming they are consistent. The result was announced by Paul Bernays in 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) Rieger-Bernays permutation models (or method), which were used for other proofs of independence for non-well-founded systems.
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Axiom of regularity
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Regularity and Russell's paradox
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Regularity and Russell's paradox
Naive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent due to Russell's paradox. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker axiom schema of separation. However, this step alone takes one to theories of sets which are considered too weak. So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension. So far, these axioms do not seem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties. These two axioms are known to be relatively consistent.
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no set of all sets. The axiom of regularity together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.
If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as Russell's paradox) which followed from the original theory would still follow in the extended theory.
The existence of Quine atoms (sets that satisfy the formula equation x = {x}, i.e. have themselves as their only elements) is consistent with the theory obtained by removing the axiom of regularity from ZFC. Various non-wellfounded set theories allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.
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Axiom of regularity
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Regularity, the cumulative hierarchy, and types
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Regularity, the cumulative hierarchy, and types
In ZF it can be proven that the class , called the von Neumann universe, is equal to the class of all sets. This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which does not satisfy the axiom of regularity, a model which satisfies it can be constructed by taking only sets in .
Herbert Enderton wrote that "The idea of rank is a descendant of Russell's concept of type". Comparing ZF with type theory, Alasdair Urquhart wrote that "Zermelo's system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included.The details of this implicit typing are spelled out in , and again in .
Dana Scott went further and claimed that:
In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity.
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Axiom of regularity
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History
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History
The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff.cf. and . Mirimanoff called a set x "regular" () if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ( in Mirimanoff's terminology).
Skolem and von Neumann pointed out that non-well-founded sets are superfluous and in the same publication von Neumann gives an axiom which excludes some, but not all, non-well-founded sets. In a subsequent publication, von Neumann gave an equivalent but more complex version of the axiom of class foundation:cf. and
The contemporary and final form of the axiom is due to Zermelo.
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Axiom of regularity
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Regularity in the presence of urelements
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Regularity in the presence of urelements
Urelements are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such as ZFA, there are. In these theories, the axiom of regularity must be modified. The statement "" needs to be replaced with a statement that is not empty and is not an urelement. One suitable replacement is , which states that x is inhabited.
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Axiom of regularity
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See also
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See also
Non-well-founded set theory
Scott's trick
Epsilon-induction
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Axiom of regularity
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References
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References
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Axiom of regularity
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Sources
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Sources
Reprinted in
Reprinted in From Frege to Gödel, van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp. 291–301.
Translation in
Translation in
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Axiom of regularity
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External links
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External links
Inhabited set and the axiom of foundation on nLab
Category:Axioms of set theory
Category:Wellfoundedness
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Axiom of regularity
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Table of Content
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short description, Elementary implications of regularity, No set is an element of itself, No infinite descending sequence of sets exists, Simpler set-theoretic definition of the ordered pair, Every set has an ordinal rank, For every two sets, only one can be an element of the other, The axiom of dependent choice and no infinite descending sequence of sets implies regularity, Regularity and the rest of ZF(C) axioms, Regularity and Russell's paradox, Regularity, the cumulative hierarchy, and types, History, Regularity in the presence of urelements, See also, References, Sources, External links
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IBM AIX
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short description
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AIX (pronounced ) is a series of proprietary Unix operating systems developed and sold by IBM since 1986. The name stands for "Advanced Interactive eXecutive". Current versions are designed to work with Power ISA based server and workstation computers such as IBM's Power line.
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IBM AIX
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Background
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Background
Originally released for the IBM RT PC RISC workstation in 1986, AIX has supported a wide range of hardware platforms, including the IBM RS/6000 series and later Power and PowerPC-based systems, IBM System i, System/370 mainframes, PS/2 personal computers, and the Apple Network Server. Currently, it is supported on IBM Power Systems alongside IBM i and Linux.
AIX is based on UNIX System V with 4.3BSD-compatible extensions. It is certified to the UNIX 03 and UNIX V7 specifications of the Single UNIX Specification, beginning with AIX versions 5.3 and 7.2 TL5, respectively. Older versions were certified to the UNIX 95 and UNIX 98 specifications.
AIX was the first operating system to implement a journaling file system. IBM has continuously enhanced the software with features such as processor, disk, and network virtualization, dynamic hardware resource allocation (including fractional processor units), and reliability engineering concepts derived from its mainframe designs.
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IBM AIX
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History
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History
thumb|IBM RS/6000 AIX file servers used for IBM.com in the 1990s
thumb|AIX Version 4 console login prompt
Unix began in the early 1970s at AT&T's Bell Labs research center, running on DEC minicomputers. By 1976, the operating system was used in various academic institutions, including Princeton University, where Tom Lyon and others ported it to the S/370 to run as a guest OS under VM/370. This port became Amdahl UTS from IBM's mainframe rival.
IBM's involvement with Unix began in 1979 when it assisted Bell Labs in porting Unix to the S/370 platform to be used as a build host for the 5ESS switch's software. During this process, IBM made modifications to the TSS/370 Resident Supervisor to better support Unix.
In 1984, IBM introduced its own Unix variant for the S/370 platform called VM/IX, developed by Interactive Systems Corporation using Unix System III. However, VM/IX was only available as a PRPQ (Programming Request for Price Quotation) and was not a General Availability product.
It was replaced in 1985 by IBM IX/370, a fully supported product based on AT&T's Unix System V, intended to compete against UTS.
In 1986, IBM introduced AIX Version 1 for the IBM RT PC workstation. It was based on UNIX System V Releases 1 and 2, incorporating source code from 4.2 and 4.3 BSD UNIX.
AIX Version 2 followed in 1987 for the RT PC.
In 1990, AIX Version 3 was released for the POWER-based RS/6000 platform. It became the primary operating system for the RS/6000 series, which was later renamed IBM eServer pSeries, IBM System p, and finally IBM Power Systems.
AIX Version 4, introduced in 1994, added symmetric multiprocessing and evolved through the 1990s, culminating with AIX 4.3.3 in 1999. A modified version of Version 4.1 was also used as the standard OS for the Apple Network Server line by Apple Computer.
In the late 1990s, under Project Monterey, IBM and the Santa Cruz Operation attempted to integrate AIX and UnixWare into a multiplatform Unix for Intel IA-64 architecture. The project was discontinued in 2002 after limited commercial success.
In 2003, the SCO Group filed a lawsuit against IBM, alleging misappropriation of UNIX System V source code in AIX. The case was resolved in 2010 when a jury ruled that Novell owned the rights to Unix, not SCO.
thumb|upright=0.75|Old logo
AIX 6 was announced in May 2007 and became generally available on November 9, 2007. Key features included role-based access control, workload partitions, and Live Partition Mobility.
AIX 7.1 was released in September 2010 with enhancements such as Cluster Aware AIX and support for large-scale memory and real-time application requirements.
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IBM AIX
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Supported hardware platforms
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Supported hardware platforms
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IBM AIX
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IBM RT PC
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IBM RT PC
The original AIX (sometimes called AIX/RT) was developed for the IBM RT PC workstation by IBM in conjunction with Interactive Systems Corporation, who had previously ported UNIX System III to the IBM PC for IBM as PC/IX. According to its developers, the AIX source (for this initial version) consisted of one million lines of code. Installation media consisted of eight 1.2M floppy disks. The RT was based on the IBM ROMP microprocessor, the first commercial RISC chip. This was based on a design pioneered at IBM Research (the IBM 801).
One of the novel aspects of the RT design was the use of a microkernel, called Virtual Resource Manager (VRM). The keyboard, mouse, display, disk drives and network were all controlled by a microkernel. One could "hotkey" from one operating system to the next using the Alt-Tab key combination. Each OS in turn would get possession of the keyboard, mouse and display. Besides AIX v2, the PICK OS also included this microkernel.
Much of the AIX v2 kernel was written in the PL.8 programming language, which proved troublesome during the migration to AIX v3. AIX v2 included full TCP/IP networking, as well as SNA and two networking file systems: NFS, licensed from Sun Microsystems, and Distributed Services (DS). DS had the distinction of being built on top of SNA, and thereby being fully compatible with DS on and on midrange systems running OS/400 through IBM i. For the graphical user interfaces, AIX v2 came with the X10R3 and later the X10R4 and X11 versions of the X Window System from MIT, together with the Athena widget set. Compilers for Fortran and C were available.
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IBM AIX
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IBM PS/2 series
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IBM PS/2 series
thumb|AIX PS/2 1.3 console login|alt=AIX PS/2 1.3 console login
AIX PS/2 (also known as AIX/386) was developed by Locus Computing Corporation under contract to IBM. AIX PS/2, first released in October 1988, ran on IBM PS/2 personal computers with Intel 386 and compatible processors.
thumb|AIX PS/2 1.3 AIXwindows Desktop|alt=AIX PS/2 1.3 AIXwindows Desktop
The product was announced in September 1988 with a baseline tag price of $595, although some utilities, such as UUCP, were included in a separate Extension package priced at $250. nroff and troff for AIX were also sold separately in a Text Formatting System package priced at $200. The TCP/IP stack for AIX PS/2 retailed for another $300. The X Window System package was priced at $195, and featured a graphical environment called the AIXwindows Desktop, based on IXI's X.desktop. The C and FORTRAN compilers each had a price tag of $275. Locus also made available their DOS Merge virtual machine environment for AIX, which could run MS DOS 3.3 applications inside AIX; DOS Merge was sold separately for another $250. IBM also offered a $150 AIX PS/2 DOS Server Program, which provided file server and print server services for client computers running PC DOS 3.3.
The last version of PS/2 AIX is 1.3. It was released in 1992 and announced to add support for non-IBM (non-microchannel) computers as well. Support for PS/2 AIX ended in March 1995.
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IBM AIX
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{{anchor
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IBM mainframes
In 1988, IBM announced AIX/370, also developed by Locus Computing. AIX/370 was IBM's fourth attempt to offer Unix-like functionality for their mainframe line, specifically the System/370 (the prior versions were a TSS/370-based Unix system developed jointly with AT&T c.1980, a VM/370-based system named VM/IX developed jointly with Interactive Systems Corporation c.1984, and a VM/370-based version of TSS/370 named IX/370 which was upgraded to be compatible with UNIX System V). AIX/370 was released in 1990 with functional equivalence to System V Release 2 and 4.3BSD as well as IBM enhancements. With the introduction of the ESA/390 architecture, AIX/370 was replaced by AIX/ESA in 1991, which was based on OSF/1, and also ran on the System/390 platform. Unlike AIX/370, AIX/ESA ran both natively as the host operating system, and as a guest under VM. AIX/ESA, while technically advanced, had little commercial success, partially because UNIX functionality was added as an option to the existing mainframe operating system, MVS, as MVS/ESA SP Version 4 Release 3 OpenEdition in 1994, and continued as an integral part of MVS/ESA SP Version 5, OS/390 and z/OS, with the name eventually changing from OpenEdition to Unix System Services. IBM also provided OpenEdition in VM/ESA Version 2 through z/VM.
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IBM AIX
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IA-64 systems
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IA-64 systems
As part of Project Monterey, IBM released a beta test version of AIX 5L for the IA-64 (Itanium) architecture in 2001, but this never became an official product due to lack of interest.
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IBM AIX
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Apple Network Servers
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Apple Network Servers
The Apple Network Server (ANS) systems were PowerPC-based systems designed by Apple Computer to have numerous high-end features that standard Apple hardware did not have, including swappable hard drives, redundant power supplies, and external monitoring capability. These systems were more or less based on the Power Macintosh hardware available at the time but were designed to use AIX (versions 4.1.4 or 4.1.5) as their native operating system in a specialized version specific to the ANS called AIX for Apple Network Servers.
AIX was only compatible with the Network Servers and was not ported to standard Power Macintosh hardware. It should not be confused with A/UX, Apple's earlier version of Unix for 68k-based Macintoshes.
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IBM AIX
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POWER ISA/PowerPC/Power ISA-based systems
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POWER ISA/PowerPC/Power ISA-based systems
thumb|AIX RS/6000 servers running IBM.com in early 1998
thumb|AIX RS/6000 servers running IBM.com in early 1998
The release of AIX version 3 (sometimes called AIX/6000) coincided with the announcement of the first POWER1-based IBM RS/6000 models in 1990.
AIX v3 innovated in several ways on the software side. It was the first operating system to introduce the idea of a journaling file system, JFS, which allowed for fast boot times by avoiding the need to ensure the consistency of the file systems on disks (see fsck) on every reboot. Another innovation was shared libraries which avoid the need for static linking from an application to the libraries it used. The resulting smaller binaries used less of the hardware RAM to run, and used less disk space to install. Besides improving performance, it was a boon to developers: executable binaries could be in the tens of kilobytes instead of a megabyte for an executable statically linked to the C library. AIX v3 also scrapped the microkernel of AIX v2, a contentious move that resulted in v3 containing no PL.8 code and being somewhat more "pure" than v2.
Other notable subsystems included:
IRIS GL, a 3D rendering library, the progenitor of OpenGL. IRIS GL was licensed by IBM from SGI in 1987, then still a fairly small company, which had sold only a few thousand machines at the time. SGI also provided the low-end graphics card for the RS/6000, capable of drawing 20,000 gouraud-shaded triangles per second. The high-end graphics card was designed by IBM, a follow-on to the mainframe-attached IBM 5080, capable of rendering 990,000 vectors per second.
PHIGS, another 3D rendering API, popular in automotive CAD/CAM circles, and at the core of CATIA.
Full implementation of version 11 of the X Window System, together with Motif as the recommended widget toolkit and window manager.
Network file systems: NFS from Sun; AFS, the Andrew File System; and DFS, the Distributed File System.
NCS, the Network Computing System, licensed from Apollo Computer (later acquired by HP).
DPS on-screen display system. This was notable as a "plan B" in case the X11+Motif combination failed in the marketplace. However, it was highly proprietary, supported only by Sun, NeXT, and IBM. This cemented its failure in the marketplace in the face of the open systems challenge of X11+Motif and its lack of 3D capability.
In addition, AIX applications can run in the PASE subsystem under IBM i.
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IBM AIX
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Source code
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Source code
IBM formerly made the AIX for RS/6000 source code available to customers for a fee; in 1991, IBM customers could order the AIX 3.0 source code for a one-time charge of US$60,000; subsequently, IBM released the AIX 3.1 source code in 1992, and AIX 3.2 in 1993. These source code distributions excluded certain files (authored by third-parties) which IBM did not have rights to redistribute, and also excluded layered products such as the MS-DOS emulator and the C compiler. Furthermore, in order to be able to license the AIX source code, the customer first had to procure source code license agreements with AT&T and the University of California, Berkeley.
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IBM AIX
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Versions
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Versions
thumb|alt=AIX 5.3 welcome banner|The default login banner for AIX 5.3 on PowerPC
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IBM AIX
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POWER/PowerPC/Power ISA releases
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POWER/PowerPC/Power ISA releases
Version Release date End of support date
AIX V7.3, December 10, 2021
Requires POWER8 or newer CPUs
AIX V7.2, October 5, 2015
Live update for Interim Fixes, Service Packs and Technology Levels replaces the entire AIX kernel without impacting applications
Flash based filesystem caching
Cluster Aware AIX automation with repository replacement mechanism
SRIOV-backed VNIC, or dedicated VNIC virtualized network adapter support
RDSv3 over RoCE adds support of the Oracle RDSv3 protocol over the Mellanox Connect RoCE adapters
Supports secure boot on POWER9 systems.
Requires POWER7 or newer CPUs
AIX V7.1, September 10, 2010
Support for 256 cores / 1024 threads in a single LPAR
The ability to run AIX V5.2 or V5.3 inside of a Workload Partition
An XML profile based system configuration management utility
Support for export of Fibre Channel adapters to WPARs
VIOS disk support in a WPAR
Cluster Aware AIX
AIX Event infrastructure
Role-based access control (RBAC) with domain support for multi-tenant environments
Requires POWER4 or newer CPUs
AIX V6.1, November 9, 2007
Workload Partitions (WPARs) operating system-level virtualization
Live Application Mobility
Live Partition Mobility
Security
Role Based Access Control RBAC
AIX Security Expert a system and network security hardening tool
Encrypting JFS2 filesystem
Trusted AIX
Trusted Execution
Integrated Electronic Service Agent for auto error reporting
Concurrent Kernel Maintenance
Kernel exploitation of POWER6 storage keys
ProbeVue dynamic tracing
Systems Director Console for AIX
Integrated filesystem snapshot
Requires POWER4 or newer CPUs
AIX 6 withdrawn from Marketing effective April 2016 and from Support effective April 2017
AIX 5L 5.3, August 13, 2004, end of support April 30, 2012
NFS Version 4
Advanced Accounting
Virtual SCSI
Virtual Ethernet
Exploitation of Simultaneous multithreading (SMT)
Micro-Partitioning enablement
POWER5 exploitation
JFS2 quotas
Ability to shrink a JFS2 filesystem
Kernel scheduler has been enhanced to dynamically increase and decrease the use of virtual processors.
AIX 5L 5.2, October 18, 2002, end of support April 30, 2009
Ability to run on the IBM BladeCenter JS20 with the PowerPC 970
Minimum level required for POWER5 hardware
MPIO for Fibre Channel disks
iSCSI Initiator software
Participation in Dynamic LPAR
Concurrent I/O (CIO) feature introduced for JFS2 released in Maintenance Level 01 in May 2003
AIX 5L 5.1, May 4, 2001, end of support April 1, 2006
Ability to run on an IA-64 architecture processor, although this never went beyond beta.
Minimum level required for POWER4 hardware and the last release that worked on the Micro Channel architecture
64-bit kernel, installed but not activated by default
JFS2
Ability to run in a Logical Partition on POWER4
The L stands for Linux affinity
Trusted Computing Base (TCB)
Support for mirroring with striping
AIX 4.3.3, September 17, 1999
Online backup function
Workload Manager (WLM)
Introduction of topas utility
AIX 4.3.2, October 23, 1998
AIX 4.3.1, April 24, 1998
First TCSEC security evaluation, completed December 18, 1998
AIX 4.3, October 31, 1997
Ability to run on 64-bit architecture CPUs
IPv6
Web-based System Manager
AIX 4.2.1, April 25, 1997
NFS Version 3
Y2K-compliant
AIX 4.2, May 17, 1996
AIX 4.1.5, November 8, 1996
AIX 4.1.4, October 20, 1995
AIX 4.1.3, July 7, 1995
CDE 1.0 became the default GUI environment, replacing the AIXwindows Desktop.
AIX 4.1.1, October 28, 1994
AIX 4.1, August 12, 1994
AIX Ultimedia Services introduced (multimedia drivers and applications)
AIX 4.0, 1994
Run on RS/6000 systems with PowerPC processors and PCI busses.
AIX 3.2.5, October 15, 1993
AIX 3.2 1992
AIX 3.1, (General Availability) February 1990
Journaled File System (JFS) filesystem type
AIXwindows Desktop (based on X.desktop from IXI Limited)
AIX 3.0 1989 (Early Access)
LVM (Logical Volume Manager) was incorporated into OSF/1, and in 1995 for HP-UX, and the Linux LVM implementation is similar to the HP-UX LVM implementation.
SMIT was introduced.
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IBM AIX
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IBM System/370 releases
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IBM System/370 releases
AIX/ESA Version 2 Release 2
Announced December 15, 1992
Available February 26, 1993
Withdrawn Jun 19, 1993
Runs only in S/370-ESA mode
AIX/ESA Version 2 Release 1
Announced March 31, 1992
Available June 26, 1992
Withdrawn Jun 19, 1993
Runs only in S/370-ESA mode
AIX/370 Version 1 Release 2.1
Announced February 5, 1991
Available February February 22, 1991
Withdrawn December 31, 1992
Does not run in XA, ESA or z mode
AIX/370 Version 1 Release 1
Announced March 15, 1988
Available February 16, 1989
Does not run in XA, ESA or z mode
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IBM AIX
|
IBM PS/2 releases
|
IBM PS/2 releases
AIX PS/2 v1.3, October 1992
Withdrawn from sale in US, March 1995
Patches supporting IBM ThinkPad 750C family of notebook computers, 1994
Patches supporting non PS/2 hardware and systems, 1993
AIX PS/2 v1.2.1, May 1991
AIX PS/2 v1.2, March 1990
AIX PS/2 v1.1, March 1989
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IBM AIX
|
IBM RT releases
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IBM RT releases
AIX RT v2.2.1, March 1991
AIX RT v2.2, March 1990
AIX RT v2.1, March 1989
X-Windows included on installation media
AIX RT v1.1, 1986
AIX RT v1.0, 1985
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IBM AIX
|
User interfaces
|
User interfaces
thumb|The Common Desktop Environment, AIX's default graphical user interface
The default shell was Bourne shell up to AIX version 3, but was changed to KornShell (ksh88) in version 4 for XPG4 and POSIX compliance.
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IBM AIX
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Graphical
|
Graphical
The Common Desktop Environment (CDE) is AIX's default graphical user interface. As part of Linux Affinity and the free AIX Toolbox for Linux Applications (ATLA), open-source KDE and GNOME desktops are also available.
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IBM AIX
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System Management Interface Tool
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System Management Interface Tool
thumb|The initial menu, when running in text mode
SMIT is the System Management Interface Tool for AIX. It allows a user to navigate a menu hierarchy of commands, rather than using the command line. Invocation is typically achieved with the command smit. Experienced system administrators make use of the F6 function key which generates the command line that SMIT will invoke to complete it.
SMIT also generates a log of commands that are performed in the smit.script file. The smit.script file automatically records the commands with the command flags and parameters used. The smit.script file can be used as an executable shell script to rerun system configuration tasks. SMIT also creates the smit.log file, which contains additional detailed information that can be used by programmers in extending the SMIT system.
smit and smitty refer to the same program, though smitty invokes the text-based version, while smit will invoke an X Window System based interface if possible; however, if smit determines that X Window System capabilities are not present, it will present the text-based version instead of failing. Determination of X Window System capabilities is typically performed by checking for the existence of the DISPLAY variable.
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IBM AIX
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Database
|
Database
Object Data Manager (ODM) is a database of system information integrated into AIX, analogous to the registry in Microsoft Windows. A good understanding of the ODM is essential for managing AIX systems.
Data managed in ODM is stored and maintained as objects with associated attributes. Interaction with ODM is possible via application programming interface (API) library for programs, and command-line utilities such as odmshow, odmget, odmadd, odmchange and odmdelete for shell scripts and users. SMIT and its associated AIX commands can also be used to query and modify information in the ODM. ODM is stored on disk using Berkeley DB files.
Example of information stored in the ODM database are:
Network configuration
Logical volume management configuration
Installed software information
Information for logical devices or software drivers
List of all AIX supported devices
Physical hardware devices installed and their configuration
Menus, screens and commands that SMIT uses
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IBM AIX
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See also
|
See also
AOS, IBM's educational-market port of 4.3BSD
IBM PowerHA SystemMirror (formerly HACMP)
List of Unix systems
nmon
Operating systems timeline
Service Update Management Assistant
Vital Product Data (VPD)
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IBM AIX
|
References
|
References
Category:IBM AIX
Category:Power ISA operating systems
Category:PowerPC operating systems
IBM Aix
Category:Object-oriented database management systems
Category:1986 software
Category:X86 operating systems
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IBM AIX
|
Table of Content
|
short description, Background, History, Supported hardware platforms, IBM RT PC, IBM PS/2 series, {{anchor, IA-64 systems, Apple Network Servers, POWER ISA/PowerPC/Power ISA-based systems, Source code, Versions, POWER/PowerPC/Power ISA releases, IBM System/370 releases, IBM PS/2 releases, IBM RT releases, User interfaces, Graphical, System Management Interface Tool, Database, See also, References
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AppleTalk
|
Short description
|
AppleTalk is a discontinued proprietary suite of networking protocols developed by Apple Computer for their Macintosh computers. AppleTalk includes a number of features that allow local area networks to be connected with no prior setup or the need for a centralized router or server of any sort. Connected AppleTalk-equipped systems automatically assign addresses, update the distributed namespace, and configure any required inter-networking routing.
AppleTalk was released in 1985 and was the primary protocol used by Apple devices through the 1980s and 1990s. Versions were also released for the IBM PC and compatibles and the Apple IIGS. AppleTalk support was also available in most networked printers (especially laser printers), some file servers, and a number of routers.
The rise of TCP/IP during the 1990s led to a reimplementation of most of these types of support on that protocol, and AppleTalk became unsupported as of the release of Mac OS X v10.6 in 2009. Many of AppleTalk's more advanced autoconfiguration features have since been introduced in Bonjour, while Universal Plug and Play serves similar needs.
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AppleTalk
|
History
|
History
|
AppleTalk
|
AppleNet
|
AppleNet
After the release of the Apple Lisa computer in January 1983, Apple invested considerable effort in the development of a local area networking (LAN) system for the machines. Known as AppleNet, it was based on the seminal Xerox XNS protocol stack but running on a custom 1 Mbit/s coaxial cable system rather than Xerox's 2.94 Mbit/s Ethernet. AppleNet was announced early in 1983 with a full introduction at the target price of $500 for plug-in AppleNet cards for the Lisa and the Apple II.
At that time, early LAN systems were just coming to market, including Ethernet, Token Ring, Econet, and ARCNET. This was a topic of major commercial effort at the time, dominating shows like the National Computer Conference (NCC) in Anaheim in May 1983. All of the systems were jockeying for position in the market, but even at this time, Ethernet's widespread acceptance suggested it was to become a de facto standard. It was at this show that Steve Jobs asked Gursharan Sidhu a seemingly innocuous question: "Why has networking not caught on?"
Four months later, in October, AppleNet was cancelled. At the time, they announced that "Apple realized that it's not in the business to create a networking system. We built and used AppleNet in-house, but we realized that if we had shipped it, we would have seen new standards coming up." In January, Jobs announced that they would instead be supporting IBM's Token Ring, which he expected to come out in a "few months".
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AppleTalk
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AppleBus
|
AppleBus
Through this period, Apple was deep in development of the Macintosh computer. During development, engineers had made the decision to use the Zilog 8530 serial controller chip (SCC) instead of the lower-cost and more common UART to provide serial port connections. The SCC cost about $5 more than a UART, but offered much higher speeds of up to 250 kilobits per second (or higher with additional hardware) and internally supported a number of basic networking-like protocols like IBM's Bisync.
The SCC was chosen because it would allow multiple devices to be attached to the port. Peripherals equipped with similar SCCs could communicate using the built-in protocols, interleaving their data with other peripherals on the same bus. This would eliminate the need for more ports on the back of the machine, and allowed for the elimination of expansion slots for supporting more complex devices. The initial concept was known as AppleBus, envisioning a system controlled by the host Macintosh polling "dumb" devices in a fashion similar to the modern Universal Serial Bus.
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AppleTalk
|
AppleBus networking
|
AppleBus networking
The Macintosh team had already begun work on what would become the LaserWriter and had considered a number of other options to answer the question of how to share these expensive machines and other resources. A series of memos from Bob Belleville clarified these concepts, outlining the Mac, LaserWriter, and a file server system which would become the Macintosh Office. By late 1983 it was clear that IBM's Token Ring would not be ready in time for the launch of the Mac, and might miss the launch of these other products as well. In the end, Token Ring would not ship until October 1985.
Jobs' earlier question to Sidhu had already sparked a number of ideas. When AppleNet was cancelled in October, Sidhu led an effort to develop a new networking system based on the AppleBus hardware. This new system would not have to conform to any existing preconceptions, and was designed to be worthy of the Mac – a system that was user-installable and required no configuration or fixed network addresses – in short, a true plug-and-play network. Considerable effort was needed, but by the time the Mac was released, the basic concepts had been outlined, and some of the low-level protocols were on their way to completion. Sidhu mentioned the work to Belleville only two hours after the Mac was announced.
The "new" AppleBus was announced in early 1984,AppleBus is mentioned by name in Steve Jobs' introduction of the Macintosh at the Boston Computer Society meeting in 1984. It appears just after the 7:20 mark in the video. allowing direct connection from the Mac or Lisa through a small box that is plugged into the serial port and connected via cables to the next computer upstream and downstream. Adaptors for Apple II and Apple III were also announced. Apple also announced that an AppleBus network could be attached to, and would appear to be a single node within, a Token Ring system. Details of how this would work were sketchy.
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AppleTalk
|
AppleTalk Personal Network
|
AppleTalk Personal Network
Just prior to its release in early 1985, AppleBus was renamed AppleTalk. Initially marketed as AppleTalk Personal Network, it comprised a family of network protocols and a physical layer.
The physical layer had a number of limitations, including a speed of only 230.4 kbit/s, a maximum distance of from end to end, and only 32 nodes per LAN. But as the basic hardware was built into the Mac, adding nodes only cost about $50 for the adaptor box. In comparison, Ethernet or Token Ring cards cost hundreds or thousands of dollars. Additionally, the entire networking stack required only about 6 kB of RAM, allowing it to run on any Mac.
The relatively slow speed of AppleTalk allowed further reductions in cost. Instead of using RS-422's balanced transmit and receive circuits, the AppleTalk cabling used a single common electrical ground, which limited speeds to about 500 kbit/s, but allowed one conductor to be removed. This meant that common three-conductor cables could be used for wiring. Additionally, the adaptors were designed to be "self-terminating", meaning that nodes at the end of the network could simply leave their last connector unconnected. There was no need for the wires to be connected back together into a loop, nor the need for hubs or other devices.
The system was designed for future expansion; the addressing system allowed for expansion to 255 nodes in a LAN (although only 32 could be used at that time), and by using "bridges" (which came to be known as "routers", although technically not the same) one could interconnect LANs into larger collections. "Zones" allowed devices to be addressed within a bridge-connected internet. Additionally, AppleTalk was designed from the start to allow use with any potential underlying physical link, and within a few years, the physical layer would be renamed LocalTalk, so as to differentiate it from the AppleTalk protocols.
The main advantage of AppleTalk was that it was completely maintenance-free. To join a device to a network, a user simply plugged the adaptor into the machine, then connected a cable from it to any free port on any other adaptor. The AppleTalk network stack negotiated a network address, assigned the computer a human-readable name, and compiled a list of the names and types of other machines on the network so the user could browse the devices through the Chooser. AppleTalk was so easy to use that ad hoc networks tended to appear whenever multiple Macs were in the same room. Apple would later use this in an advertisement showing a network being created between two seats in an airplane.
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AppleTalk
|
PhoneNet and other adaptors
|
PhoneNet and other adaptors
A thriving third-party market for AppleTalk devices developed over the next few years. One particularly notable example was an alternate adaptor designed by BMUG and commercialised by Farallon as PhoneNET in 1987. This was essentially a replacement for Apple's connector that had conventional phone jacks instead of Apple's round connectors. PhoneNet allowed AppleTalk networks to be connected together using normal telephone wires, and with very little extra work, could run analog phones and AppleTalk on a single four-conductor phone cable.
Other companies took advantage of the SCC's ability to read external clocks in order to support higher transmission speeds, up to 1 Mbit/s. In these systems, the external adaptor also included its own clock, and used that to signal the SCC's clock input pins. The best-known such system was Centram's FlashTalk, which ran at 768 kbit/s, and was intended to be used with their TOPS networking system. A similar solution was the 850 kbit/s DaynaTalk, which used a separate box that plugged in between the computer and a normal LocalTalk/PhoneNet box. Dayna also offered a PC expansion card that ran up to 1.7 Mbit/s when talking to other Dayna PC cards. Several other systems also existed with even higher performance, but these often required special cabling that was incompatible with LocalTalk/PhoneNet, and also required patches to the networking stack that often caused problems.
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