Semigroup

A semigroup is a nonempty set S equipped with a binary operation, usually written as a function from S × S to S, that is associative: for all a, b, c in S, (a·b)·c = a·(b·c). The operation is required to be closed, so a·b is always in S. If there exists an element e in S with e·a = a·e = a for all a in S, then S is a monoid. If every element has an inverse with respect to the operation and an identity exists, S is a group.

Common examples include the natural numbers (including zero) under addition, the integers under addition, and strings over a finite alphabet under concatenation. The set of nonzero integers under multiplication is a semigroup, though not a monoid. The set of all n×n matrices over a field under matrix multiplication forms a semigroup and is a monoid since the identity matrix serves as the identity element.

A useful general fact is Cayley’s type representation for semigroups: every semigroup S can be embedded into the full transformation semigroup on S by left multiplication, mapping each a in S to the function La(x) = a·x. This shows, in particular, that semigroups can be studied via actions on sets.

Structure theory for semigroups includes concepts such as idempotents (elements e with e·e = e) and Green’s relations, which organize elements by the ideals they generate. Semigroups can be commutative or noncommutative, finite or infinite, and they arise naturally in areas such as algebra, combinatorics, and computer science.

Applications include automata theory and the theory of regular languages, where finite semigroups provide algebraic tools for understanding language recognition and classification.