Semigruppi

Semigruppi, or semigroups, are algebraic structures consisting of a nonempty set S equipped with an associative binary operation. That is, for all a, b, c in S, (a • b) • c = a • (b • c). Unlike groups, semigruppi do not require an identity element or inverses. If an identity exists, the structure is called a monoid; if every element has an inverse, it is a group.

Semigroup theory studies the properties and relations that arise from this structure. Subsemigruppi are subsets closed under the operation. Homomorphisms preserve the operation, and isomorphisms identify semigruppi with essentially the same structure. The field also considers special classes such as monoids (semigruppi with identity), groups, and various regular or idempotent semigruppi, using tools like Green's relations and presentations by generators and relations.

Examples include the set of non-negative integers under addition, which forms a semigroup and, with 0 as identity, a monoid. The set of all finite strings over a fixed alphabet under concatenation is a semigroup; including the empty string yields a monoid, while nonempty strings form a free semigroup. Other common examples are endomorphisms of a set under composition and matrices under multiplication (when restricted to appropriate sets).

Applications of semigruppi span computer science, particularly automata theory and formal languages, where concatenation of symbols models sequence processing. They also appear in combinatorics and algebra, where semigroup algebras provide a bridge to representation theory and geometry. Semigruppi thus offer a flexible framework for modeling composition-like operations without requiring inverses.