By definition, a semigroup is an associative magma. A semigroup with an identity element is called a monoid. A group is then a monoid in which every element has an inverse element. Semigroups must not be confused with quasigroups, which are sets with a not necessarily associative binary operation such that division is always possible. More succinc...
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http://en.wikipedia.org/wiki/Semigroup
(from the article `automata theory`) A mathematically significant classification of transducers may be obtained in terms of the theory of semi-groups. In outline, if the transducer is ...
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http://www.britannica.com/eb/a-z/s/63
A set together with a method of combining elements, such as addition or multiplication, to get new ones, which satisfies only some of the properties required to get a group. In particular, a semigroup need not have an identity element and elements need not have inverses.
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http://www.daviddarling.info/encyclopedia/S/semigroup.html
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