Zappa–Szép products of semigroups provide a rich class
of examples of semigroups that include the self-similar
group actions of Nekrashevych. We use Li’s construction of
semigroup C∗-algebras to associate a C∗-algebra to Zappa–
Szép products and give an explicit presentation of the algebra.
We then define a quotient C∗-algebra that generalises the
Cuntz–Pimsner algebras for self-similar actions. We indicate
how known examples, previously viewed as distinct classes,
fit into our unifying framework. We specifically discuss the
Baumslag–Solitar groups, the binary adding machine, the
semigroup NN×, and the ax + b-semigroup Z Z×.