Firat Ateş

Gröbner-Shirshov bases of some monoids

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

Gröbner-Shirshov Bases of the Generalized Bruck-Reilly ∗-Extension

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

Knit Products of Some Groups and Their Applications

Let G be a group with subgroups A and K (not necessarily normal) such that G = AK and A ∩ K = {1}. 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. 2000 Mathematics Subject Classification: 20E22, 20F05, 20F55, 20F32.

A new monoid construction under crossed products

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.

Groups St Andrews 2005: Minimal but inefficient presentations for semi-direct products of finite cyclic monoids

Let A and K be arbitrary two monoids. For any connecting monoid homomorphism θ : A −→ End(K), let M = K oθ A be the corresponding monoid semi-direct product. In [3], Cevik discussed necessary and sufficient conditions for the standard presentation of M to be efficient (or, equivalently, p-Cockcroft for any prime p or 0), and then, as an application of this, he showed the efficiency for the presentation, say PM , of the semi-direct product of any two finite cyclic monoids. As a main result of this paper, we give sufficient conditions for PM to be minimal but not efficient. To do that we will use the same method as given in [4]. 2000 Mathematics Subject Classification: 20L05, 20M05, 20M15, 20M50, 20M99.

On the Norms of Toeplitz and Hankel Matrices with Pell Numbers

Let us define A = [a ij ] i,i = 0 n-1 and B = [b ij ] i,i = 0 n-1 as n×n Toeplitz and Hankel matrices, respectively, such that a ij = P i−j and b ij = P i+j , where P denotes the Pell number. We present upper and lower bounds for the spectral norms of these matrices.

Cyclic) subgroup separability of HNN and split extensions

This work has been divided in two parts. In the first part we give a sufficient conditions on an HNN extension of a free group to be cyclic subgroup seperable. In the second part we define just subgroup separability on a split extension of special groups which is actually on holomorph.

The graph based on Gröbner-Shirshov bases of groups

Let us consider groups G1=Zk∗(Zm∗Zn), G2=Zk×(Zm∗Zn), G3=Zk∗(Zm×Zn), G4=(Zk∗Zl)∗(Zm∗Zn) and G5=(Zk∗Zl)×(Zm∗Zn), where k,l,m,n≥2. In this paper, by defining a new graph Γ(Gi) based on the Gröbner-Shirshov bases over groups Gi, where 1≤i≤5, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of Γ(Gi). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in such fields as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. In addition, the Gröbner-Shirshov basis and the presentations of algebraic structures contain a mixture of algebra, topology and geometry within the purposes of this journal.MSC:05C25, 13P10, 20M05, 20E06, 26C10.

The Efficiency of the Semi‐Direct Products of Free Abelian Monoid with Rank n by the Infinite Cyclic Monoid

In this paper we give necessary and sufficient conditions for the efficiency of the semi‐direct product of free abelian monoid with rank n by the infinite cyclic monoid.

Conjugacy for Free Groups under Split Extensions

At the present paper we show that conjugacy is preserved and reflected by the natural homomorphism defined from a semigroup S to a group G, where G defines split extensions of some free groups. The main idea in the proofs is based on a geometrical structure as applied in the paper [8].