Join and meet

In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) of S, denoted ⋁S, and infimum (greatest lower bound) of S, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of P.

Join and meet can also be defined as a commutative, associative and idempotent partial binary operation on pairs of elements from P. If a and b are elements from P, the join is denoted as a ∨ b and the meet is denoted a ∧ b.

Join and meet are symmetric duals with respect to order inversion. The join/meet of a subset of a totally ordered set is simply its maximal/minimal element.

A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.

! (disambiguation)

! is a punctuation mark called an exclamation mark (33 in ASCII), exclamation point, ecphoneme, or bang.

! may also refer to:

Mathematics and computers

Music

Other

See also

Junior Certificate

The Junior Certificate (Irish: Teastas Sóisearach) is an educational qualification awarded in Ireland by the Department of Education and Skills to students who have successfully completed the junior cycle of secondary education, and achieved a minimum standard in their Junior Certification examinations. These exams, like those for the Leaving Certificate, are supervised by the State Examinations Commission. A "recognised pupil"<ref name"">Definitions, Rules and Programme for Secondary Education, Department of Education, Ireland, 2004</ref> who commences the Junior Cycle must reach at least 12 years of age on 1 January of the school year of admission and must have completed primary education; the examination is normally taken after three years' study in a secondary school. Typically a student takes 9 to 13 subjects – including English, Irish and Mathematics – as part of the Junior Cycle. The examination does not reach the standards for college or university entrance; instead a school leaver in Ireland will typically take the Leaving Certificate Examination two or three years after completion of the Junior Certificate to reach that standard.

The Servant (band)

The Servant was an English alternative band, formed in London in 1998. They are popular in France, Spain, Switzerland as well as other European countries.

Their first introduction to an American audience was in the trailer of the film Sin City with the instrumental version of their song "Cells". This version of "Cells" is not on the Sin City soundtrack, but it can be downloaded via their website ("Cells" was also used in the film The Transporter and Transporter 2, along with their song, "Body"). Since the Sin City trailers, there has been significant U.S. interest in their records and demands for live concerts. The band released their fourth album entitled How To Destroy A Relationship in 2006.

Before achieving commercial success in 2004 with their self-titled album, released by Prolifica Records in the UK and by Recall Group in France, The Servant released two EP's: Mathematics in 1999 and With the Invisible in 2000.

On 26 November 2007, the band announced on their blog at MySpace that they were splitting up "to move on to pastures new".

Join

Join may refer to:

Lattice (order)

In mathematics, a lattice is one of the fundamental algebraic structures used in abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.

Lattices as partially ordered sets

If (L, ≤) is a partially ordered set (poset), and S⊆L is an arbitrary subset, then an element u∈L is said to be an upper bound of S if s≤u for each s∈S. A set may have many upper bounds, or none at all. An upper bound u of S is said to be its least upper bound, or join, or supremum, if u≤x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one. Dually, l∈L is said to be a lower bound of S if l≤s for each s∈S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, if x≤l for each lower bound x of S. A set may have many lower bounds, or none at all, but can have at most one greatest lower bound.

Join (topology)

In topology, a field of mathematics, the join of two topological spaces A and B, often denoted by , is defined to be the quotient space

where I is the interval [0, 1] and R is the equivalence relation generated by

At the endpoints, this collapses to and to .

Intuitively, is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in A to every point in B.

Properties

is homeomorphic to the reduced suspension

of the smash product. Consequently, since is contractible, there is a homotopy equivalence

Examples