In this paper we first define a presentation for the generalized Bruck-Reilly ∗-extension of a monoid and then we work on its Gröbner-Shirshov bases.

The main goal of this paper is to define Gröbner–Shirshov bases for some monoids.\ud Therefore, after giving some preliminary material, we first give Gröbner–Shirshov bases for\ud graphs and Schützenberger products of monoids in separate sections. In the final section,\ud we further present a Gröbner–Shirshov basis for a Rees matrix semigroup

Let G be a group with subgroups A and K (not necessarily normal) such that G AK and A K f1g. Then G is isomorphic to the knit product, that is, the``two-sided semidirect product'' of K by A. We note that knit products coincide with Zappa-Szep products (see [18]). In this paper, as an application of [2, Lemma 3.16], we first define a presentation for the knit product G where A and K are finite cyclic subgroups. Then we give an example of this presentation by considering the (extended) Hecke groups. After that, by defining the Schur multiplier of G, we present sufficient conditions for the presentation of G to be efficient. In the final part of this paper, by examining the knit product of a free group of rank n by an infinite cyclic group, we give necessary and sufficient conditions for this special knit product to be subgroup separable.

Let G be a semi-direct product of B by A where B and A are both cyclic groups of order n (n ∈ N) and p (any prime), respectively. As a main result of this paper, we prove that G has an inefficient but minimal presentation. Then, as an application of this result, we show that a metacyclic group satisfy the main result. ÖZETB ve A nın her ikiside sırasıyla n (n ∈ N) ve p (asal) mertebeli devirli gruplar olmak üzere, G grubu B nin A ile yarı-direkt çarpımı olsun. Bu çalışmanın ana sonucu olarak G nin etkili olmayan ancak minimal olan bir sunuşa sahip olduğunu ispatladık. Daha sonra bu sonucun bir uygulaması olarak metadevirli grupların bu sonucu sağladığını gösterdik.

In this paper we define a new monoid construction under crossed products for given monoids. We also present a generating set and a relator set for this product. Finally, we give the necessary and sufficient conditions for the regularity of it. MSC: 05C10; 05C12; 05C25; 20E22; 20M05