A semigroup is an associative magma.
Let $(S,*)$ be a semigroup. $T\subseteq S$ is a subsemigroup as soon as $T$ is closed under $*$.
DEF-SEM-HOM. Semigroup Homomorphism.
A semigroup homomorphism between two semigroups $(S,*)$ and $(T,\diamond)$ is a function $f:S\to T$ such that for all $a$, $b\in M$,
\[f(a*b) = f(a)\diamond f(b).\]