In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. The binary operation of a semigroup is most denoted multiplicatively: x·y, or xy, denotes the result of applying the semigroup operation to the ordered pair. Associativity is formally expressed as that ·z = x· for all x, y and z in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses; as in the case of groups or magmas, the semigroup operation need not be commutative, so x·y is not equal to y·x. If the semigroup operation is commutative the semigroup is called a commutative semigroup or it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between groups and semigroups, is a semigroup having an identity element, thus obeying all but one of the axioms of a group. A natural example is strings with concatenation as the binary operation, the empty string as the identity element.

Restricting to non-empty strings gives an example of a semigroup, not a monoid. Positive integers with addition form a commutative semigroup, not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be turned into a monoid by just adding an identity element. Monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups in a different direction. Division in semigroups is not possible in general; the formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as transformation semigroup, in which arbitrary functions replace the role of bijections from group theory. A deep result in the classification of finite semigroups is Krohn–Rhodes theory, analagous to the Jordan–Hölder decomposition for finite groups; some other techniques for studying semigroups, like Green's relations, do not resemble anything in group theory.

The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov processes. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications; some of these classes are closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There interesting classes of semigroups that do not contain any groups except the trivial group. A semigroup is a set S together with a binary operation " ⋅ " that satisfies the associative property: For all a, b, c ∈ S, the equation ⋅ c = a ⋅ holds.

More succinctly, a semigroup is an associative magma. Empty semigroup: the empty set forms a semigroup with the empty function as the binary operation. Semigroup with one element: there is only one, the singleton with operation a · a = a. Semigroup with two elements: there are five which are different; the "flip-flop" monoid: a semigroup with three elements representing the three operations on a switch - set, do nothing. The set of positive integers with addition; the set of integers with minimum or maximum. Square nonnegative matrices of a given size with matrix multiplication. Any ideal of a ring with the multiplication of the ring; the set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation — the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. A probability distribution F together with all convolution powers of F, with convolution as the operation; this is called a convolution semigroup.

A group is a monoid. Transformation monoids; the set of continuous functions from a topological space to itself with composition of functions forms a monoid with the identity function acting as the identity. More the endomorphisms of any object of a category form a monoid under composition. A left identity of a semigroup S {

Double Whammy (film)

Double Whammy is a 2001 comedy/drama film written and directed by Tom DiCillo and starring Denis Leary, Elizabeth Hurley and Steve Buscemi. Although intended to be released in theaters, it was distributed direct-to-video. Ray Pluto has horrid memories of watching his child die in a traffic accident. He's a cop who's the laughingstock of New York City because his back went out while trying to stop a mass murderer —, shot by a child. For his back, he gets help from a chiropractor. Meanwhile, a teenager hires thugs to kill her father. In the same building, two young men are writing a movie script. Ray tries to get past his grief to solve the assault on the super and return the affections of the chiropractor. Denis Leary as Ray Pluto, a police officer suffering from back problems. Elizabeth Hurley as Dr. Ann Beamer, Pluto's chiropractor. Steve Buscemi as Jerry Cubbins, Pluto's partner. Melonie Diaz as Maribel Benitez, who hires thugs to kill her father. Luis Guzmán as Juan Benitez, Maribel's father. Donald Faison as Cletis, one of the writers in the building.

Victor Argo, Chris Noth, Keith Nobbs, Otto Sanchez Kevin Olson, Bill Boggs, Gerry Bamman, Sally Jessy Raphael, Sharon Wilkins and Caprice Benedetti star in the film in various roles. The film holds a 30% "Rotten" rating on Rotten Tomatoes, based on 10 reviews, 3 of which being positive. Double Whammy on IMDb Double Whammy at AllMovie Double Whammy at Rotten Tomatoes

USS Cubera (SS-347)

USS Cubera, a Balao-class submarine, was a ship of the United States Navy named for the cubera, a large fish of the snapper family found in the West Indies. Cubera was launched 17 June 1945 by Electric Boat Co. Groton, Conn.. After shakedown training off New London, Cubera arrived at Key West, Fla. 19 March 1946. She tested sonar equipment, provided services to experimental antisubmarine warfare development projects in the Florida Straits, joined in fleet exercises until 4 July 1947 when she sailed to Philadelphia Naval Shipyard for an extensive GUPPY II modernization. Returning to Key West 9 March 1948 Cubera continued to operate locally out of this port, as well as taking part in fleet exercises in the Caribbean and Atlantic until 3 July 1952 when she arrived at Norfolk, her new home port. Cubera appeared in Ray Harryhausen's It Came from Beneath the Sea, playing an "atomic sub" used to dispatch the film's giant octopus. Through 1957 Cubera conducted local operations, participated in fleet exercises in the Caribbean, as well as cruising to Sydney, Nova Scotia, in June 1955.

During 1959 and 1960, she was assigned to Task Force Alfa, a force conducting constant experiments to improve antisubmarine warfare techniques. With this group she cruised the western Atlantic from Nova Scotia to Bermuda. Cubera was decommissioned and sold under the Security Assistance Program to Venezuela 5 January 1972; the Venezuelan Navy renamed her ARV Tiburon. She was subsequently scrapped by Venezuela in 1989. List of ship launches in 1945 List of ship commissionings in 1945, List of ship commissionings in 1972 List of ship decommissionings in 1972, List of ship decommissionings in 1989 Photo gallery of Cubera at NavSource Naval History USS Cubera website