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(circle plus B) by B has been widely studied. After that a presentation for the Zappa product of A by B was defined which can be thought as the mutual semidirect product of given these two monoids under a homomorphism psi : A -> J(B) and an anti-homomorphism delta : B -> J(A) into the full transformation monoid on B, respectively on A. As a next step of these above results, by considering the monoids A(circle plus B) and B-circle plus A, we first introduce an extended version (generalization) of the Zappa product and then we prove the existence of an implicit presentation for this new product. Furthermore we 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 point out some possible future problems related to this subject.