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 semigroups C*-algebras to associate a C*-algebra to Zappa-Szep 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 knowne examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup NXNx, and the ax+b semigroup ZXZx.
Brownlowe, N. D., Ramagge, J., Robertson, D. I. & Whittaker, M. F. (2014). Zappa-Szep products of semigroups and their C*-algebras. Journal of Functional Analysis, 266 (6), 3937-3967.