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 ⊕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 ψ : A → T(B) and an anti-homomorphism δ : B → T(A) into the full transformation monoid on B, respectively on A. As a next step of these above results, by considering the monoids A ⊕B and B ⊕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.

The Arithmetic of Generalization for General Products of Monoids

“…These two constructions has been used by group theorists for many years, and in the last years it used widely in the study of semigroups. In [2] the authors bringing together the definitions of general product and (restricted) wreath products of monoids A ⊕B and B ⊕A and then gave a presentation and some other results of the main theories in terms of finite and infinite cases for this generalization "general products of wreath products" which can be thought as a generalization of results in [6] and [9]. We denote this generalization by A ⊕B δ ψ B ⊕A .…”

Some algebraic structures on the generalization general products of monoids and semigroups

“…In the literature, there are some key stone studies on the general product which is also referred as bilateral semidirect products (see [11]), Zappa products (see [7,12,16,18]) or knit products (see [1,14]). As a next step of general product, in [4], the same authors of this paper have recently introduced the generalization of the general product under the name of a higher version of Zappa products for monoids as in the following:…”