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
Typical examples include the inclusion map from the natural numbers under addition to the integers under addition,
The image of a homomorphism is always a subsemigroup of the target, while the preimage of a
Semigroup homomorphisms play an important role in abstract algebra, computer science (e.g., in the theory of