semigroupe

A semigroup is a nonempty set S equipped with an associative binary operation ·: S × S → S, meaning that for all a, b, c in S, (a · b) · c = a · (b · c). The defining property is closure under the operation and associativity; no identity element is required.

If there exists an element e in S such that e · a = a · e = a for every a in S, then (S, ·) is called a monoid. If, in addition, every element has an inverse with respect to the operation, the structure is a group. Not all semigroups are monoids or groups, and many semigroups lack inverses or identities altogether.

Examples illustrate the concept. The set of natural numbers with addition (N, +) is a semigroup; it is a monoid with 0 as the identity. The natural numbers with multiplication (N, ×) form a monoid with 1 as the identity. Strings over an alphabet under concatenation form a semigroup, and endomorphisms of a set under composition form a monoid. More generally, any set equipped with an associative binary operation is a semigroup.

Substructures and mappings are central in semigroup theory. A subset closed under the operation is a subsemigroup. A function f between semigroups is a homomorphism if f(a · b) = f(a) · f(b) for all a, b. Quotients can be formed via congruences, leading to a rich theory with connections to automata and formal languages.

Applications of finite semigroups appear in computer science, notably in automata theory and the study of regular languages through syntactic semigroups. Finite semigroups also play a role in decomposition theories and the understanding of algebraic properties via idempotents and Green’s relations.