Semigrupo

Semigrupo, in mathematics, is an algebraic structure consisting of a nonempty set with a single binary operation that is associative. Formally, a semigrupo (S, •) satisfies that for all a, b, c in S, (a • b) • c = a • (b • c). The operation need not have an identity element or inverses, which distinguishes semigroups from more restrictive structures.

If there exists an identity element e in S such that a • e = e • a = a for all a in S, the semigrupo is called a monoid. If every element has an inverse with respect to the operation, the structure is a group. The concepts of semigroups cover a broad class of algebraic systems, including many that arise in mathematics and computer science.

Common examples illustrate the range of semigroups. The natural numbers with addition form a commutative monoid, as does the natural numbers with multiplication, which has identity 1. The set of all strings over a fixed alphabet under concatenation is a semigroup, and, with an identity string, a monoid. The set of all n×n matrices over a field under matrix multiplication is also a monoid, with the square identity matrix as the identity element. More generally, the set of all functions from a fixed set to itself under composition is a monoid.

Variants and related notions include commutative semigroups (where the operation is ab = ba), idempotent semigroups or bands (where a • a = a for all a), and inverse semigroups, which generalize groups by relaxing the requirement of global inverses. Semigroup theory studies subsemigroups, homomorphisms, ideals, and relations such as Green’s relations, providing tools for understanding structure and representations.

Applications of semigroups appear in automata theory, formal languages, and the study of dynamical systems, where they model the composition of transformations and processes.