Semigruppihomomorfismi

Semigroup homomorphisms are structure–preserving maps between semigroups. A semigroup is a set equipped with an associative binary operation, but unlike a group it need not contain an identity element or inverses. A function \(f\colon S\to T\) between two semigroups \((S,\cdot)\) and \((T,\ast)\) is called a homomorphism if for all \(a,b\in S\) the equality \(f(a\cdot b)=f(a)\ast f(b)\) holds. This condition ensures that the image of the product equals the product of the images, thereby respecting the semigroup structure.

Because a semigroup may lack an identity element, homomorphisms do not need to preserve identities. If the semigroups are in fact monoids (semigroups with identity), a homomorphism that also maps the identity of the source to the identity of the target is called a monoid homomorphism; this additional requirement is not part of the general definition.

Typical examples include the inclusion map from the natural numbers under addition to the integers under addition, the map sending each integer to its remainder modulo \(m\), and the exponentiation map from the natural numbers under addition to a multiplicative semigroup of matrices. In each case the operation on the output set is compatible with the operation on the input set.

The image of a homomorphism is always a subsemigroup of the target, while the preimage of a subsemigroup is a subsemigroup of the source. Kernels are defined differently from group theory; rather than a normal subgroup, one considers the set of pairs \((a,b)\) with \(f(a)=f(b)\), which is a congruence relation on the semigroup.

Semigroup homomorphisms play an important role in abstract algebra, computer science (e.g., in the theory of formal languages) and category theory, where semigroups and their homomorphisms form a category that is foundational for many algebraic constructions.