In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. A bounded poset is a poset that has both a greatest element and a least element.

Formally, given a partially ordered set (P, ≤), then an element g of a subset S of P is the greatest element of S if

Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S.

Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. A finite chain always has a greatest and a least element.

Alicia Augello Cook (born January 25, 1981), better known by her stage name Alicia Keys, is an American R&B singer-songwriter, record producer, and actress. Keys was raised by a single mother in the Hell's Kitchen area of Manhattan in New York City. At age seven, Keys began playing the piano. She attended Professional Performing Arts School and graduated at 16 as valedictorian. Keys released her debut album with J Records, having had previous record deals first with Columbia and then Arista Records.

Keys' debut album, Songs in A Minor, was a commercial success, selling over 12 million copies worldwide. She became the best-selling new artist and best-selling R&B artist of 2001. The album earned Keys five Grammy Awards in 2002, including Best New Artist and Song of the Year for "Fallin'". Her second studio album, The Diary of Alicia Keys, was released in 2003 and was also another success worldwide, selling eight million copies. The album garnered her an additional four Grammy Awards in 2005. Later that year, she released her first live album, Unplugged, which debuted at number one in the United States. She became the first female to have an MTV Unplugged album to debut at number one and the highest since Nirvana in 1994.

Michael Gordon Oldfield (born 15 May 1953) is an English multi-instrumentalist musician and composer, working a style that blends progressive rock, folk, ethnic or world music, classical music, electronic music, New Age, and more recently, dance. His music is often elaborate and complex in nature. He is best known for his hit 1973 album Tubular Bells, which launched Virgin Records, and for his 1983 hit single "Moonlight Shadow". He is also well known for his hit rendition of the Christmas piece, "In Dulci Jubilo".

Oldfield's parents are Raymond Oldfield, a general practitioner, and his wife Maureen Liston, a nurse. His sister Sally and brother Terry are successful musicians in their own right and have appeared on several of his albums. Mike Oldfield was born in the Battle Hospital, Reading, Berkshire, and he attended St Joseph's Convent School, Highlands Junior School in Tilehurst, St Edward's Preparatory School, and Presentation College in Reading. When he was 13 he moved with his parents to Harold Wood in northeast London, where he attended Hornchurch Grammar School. He only stayed there long enough to take one GCE examination, in English, as he had already begun his music career.