Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.