A Higher Version of Zappa Products for Monoids

Abstract

For arbitrary monoids $A$ and $B$, a presentation for the restricted wreath product of $A$ by $B$ that is known as the semi-direct product of $A^{\oplus B}$ by $B$ has been widely studied (see, for instance, \cite{Cevik,Howie-Ruskuc,Johnson,Meldrum}). After that a presentation for the Zappa product of $A$ by $B$ is defined (\cite{Brin,Lavers}) which can be thought as the mutual semidirect product of given these two monoids under a homomorphism $\psi : A \rightarrow \tau(B)$ and an anti-homomorphism $\delta : B \rightarrow \tau(A)$ into the full transformation monoid on $B$, resp. on $A$ (cf. \cite{Kunze}). As a next step of these above results, by considering the monoids $A^{\oplus B}$ and $B^{\oplus A}$, we will first introduce an extended version (generalization) of the Zappa product and then we will prove the existence of an implicit presentation for this new product. Furthermore we will present some other outcomes of the main theories in terms of finite and infinite cases, and also in terms of groups. At the final part of this paper we will point out some possible future problems related to this subject.