Semigroup

A semigroup is a set equipped with an associative binary operation. The operation combines any two elements of the set, and regrouping a product does not change its value.

Binary operation: A rule that assigns an element of a set to each ordered pair of elements from that set. A semigroup’s operation must combine every pair of its elements.
Associative algebra: An algebra over a field whose multiplication is associative. Its multiplication gives a semigroup on the algebra’s elements.
Transformation semigroup: A semigroup of functions from a set to itself, with function composition as the operation. Function composition gives a direct model of associative multiplication.
Monoid: A semigroup with an identity element. A monoid is precisely a semigroup with a neutral element.
Arthur Cayley: A British mathematician who helped establish abstract algebra and introduced the group representation now called Cayley’s theorem. His transformation-based algebraic viewpoint anticipates semigroup representations.
Associativity: A property of a binary operation in which regrouping three inputs does not change the result. Associativity is the defining law that distinguishes semigroups from arbitrary binary systems.
Word (mathematics): A finite sequence of symbols drawn from a specified alphabet. Concatenation of words is an associative operation, producing a standard free semigroup.
Transition system: A model of states connected by possible actions or transitions. Sequences of actions compose associatively and can be represented by semigroup elements.
Commutative semigroup: A semigroup whose operation gives the same result when its inputs are swapped. Commutativity is an optional law beyond semigroup associativity.
Anton Suschkewitsch: A mathematician known for early structural results on finite semigroups. His work helped establish the study of finite semigroup structure.

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