Zappa-Szep products of semigroups and their C*-algebras

<p>Zappa-Szep 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 <i>C</i>*-algebras to associate a <i>C</i>*-algebra to Zappa-Szep products and give an explicit presentation of the algebra. We then define a quotient <i>C</i>*-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 N x N^×, and the ax + b-semigroup Z x Z^×.