A semigroup $G$ is a set together with a binary operation $\cdot: G \times G \longrightarrow G$ which satisfies the associative property: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in G$

The set $G$ is not required to be nonempty.

Let $G,H$ be two semigroups. A semigroup homomorphism from $G$ to $H$ is a function $f:G\to H$ such that $f(ab)=f(a)f(b)$