Monogenic Semigroups Research Articles

ON LEAP ZAGREB INDICES OF A SPECIAL GRAPH OBTAINED BY SEMIGROUPS

In 2013, Das et al. defined the monogenic semigroup graphs [10]. And, various topological indices of the monogenic semigroup graphs have been calculated so far [3,21]. The aim of this study is to continue to create formulas for the topological indices of these special graphs. In this study, we give exact formulae for various the leap Zagreb indices of this special algebraic graph obtained from monogenic semigroups.

Sum Connectivity Index Under the Cartesian and Strong Products Graph of Monogenic Semigroup

This field’s main feature is to implement the sum connectivity index method. This sum connectivity index method can solve the monogenic semigroups under the cartesian and strong products. We will define for an undirected graph as SCI(GMS)=Σuv∈E(GMS) [dGMS(u)+dGMS(v)]−1/2, where dGMS(u) and dGMS(v) are degree of u and v in GMS respectively. Further, we investigate two different algorithms concerning topological index for computing cartesian and strong products of a monogenic semigroup with a detailed example.

On the power graph of a monogenic semigroup

Let [Formula: see text] be a monogenic semigroup with zero. Here, we consider the power graph [Formula: see text] over [Formula: see text] with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] divides [Formula: see text] or [Formula: see text] divides [Formula: see text], where [Formula: see text]. We compute various graph parameters of [Formula: see text] and topological indices based on distance of vertices. Finally, we compute some graph parameters of the cartesian product [Formula: see text] of graphs [Formula: see text] and [Formula: see text].

On the clique number and independence number of the cyclic graph of a semigroup

The cyclic graph [Formula: see text] of a semigroup [Formula: see text] is the simple undirected graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] are adjacent if the subsemigroup generated by [Formula: see text] and [Formula: see text] is monogenic. In this paper, we determine the clique number of [Formula: see text] for an arbitrary semigroup [Formula: see text]. Further, we obtain the independence number of [Formula: see text] if [Formula: see text] is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of [Formula: see text] if [Formula: see text] is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.

A new authentication scheme based on monogenic semigroup

The reduction of constrained mathematical structures leads us to generalize any abstract structures. Using minimum conditions to construct a secure and robust component of the modern encryption algorithm is one crucial problem in multimedia security. With this understanding, we have proposed a new algebraic structure, namely monogenic semigroup, to construct a digital information authentication scheme. Authentication is always completed at the beginning of the application, before any throttling or approval checks are performed, and before any other code is allowed to begin running in the background. Many authentication schemes offer a complex structure for implementation in cryptosystems and applications. The anticipated mechanism uses a simple mathematical structure having the least conditions as compared to other mathematical structures. The suggested scheme provides structures for the authentication of text messages and images.

On Vertex-Degree-Based Indices of Monogenic Semigroup Graphs

Albertson and the reduced Sombor indices are vertex-degree-based graph invariants that given in [5] and [18], defined as
 
 Alb(G)=\sum_{uv\in E(G)}\left|d_{u}-d_{v}\right|, SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{u}-1)^{2}+(d_{v}-1)^{2}},
 
 respectively. 
 
 In this work we show that a calculation of Albertson and reduced Sombor index which are vertex-degree-based topological indices, over monogenic semigroup graphs.

Sombor Index over the Tensor and Cartesian Products of Monogenic Semigroup Graphs

Consider a simple graph G with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G, which is invariant under the symmetry of G. The Sombor index of G is a new graph invariant defined as SO(G)=∑uv∈E(G)(du)2+(dv)2. In this work, we connected the theory of the Sombor index with abstract algebra. We computed this topological index over the tensor and Cartesian products of a monogenic semigroup graph by presenting two different algorithms; the obtained results are illustrated by examples.

An Application of Sombor Index over a Special Class of Semigroup Graph

Recently, Gutman introduced a class of novel topological invariants named Sombor index which is defined asSOG=∑uv∈EGdu2+dv2. In this study, the Sombor index of monogenic semigroup graphs, which is an important class of algebraic structures, is calculated.

On Join Operation of Graphs by Obtained Monogenic Semigroups

For each commutative ring $R$ we associate a simple graph $\Gamma(R)$. This relationship presents a link between algebra and graph theory. Our main scope in this study is to extend this study over the special algebraic graphs to join graph operations. In this paper, we will give some graph parameters for the join of monogenic semigroup graphs.

Line Graphs of Monogenic Semigroup Graphs

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

The Tensor Product of Zero-Divisor Graphs of Variation Monogenic Semigroups

The tensor product of zero-divisor graphs of variation monogenic semigroups Γ(VS_Mn^1) and Γ(VS_Mm^2) is studied. The vertices(x_1^i,x_2^j) and (x_1^k,x_2^f) of the tensor product of this graph are adjacent whenever gcd(i,k)=1,i+k>n,gcd(j,f)=1 ,j+f>m. Some properties of tensor product graphs are obtained, such as girth, diameter, chromatic, clique and domination numbers.

On the Wiener Index of the Dot Product Graph over Monogenic Semigroups

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the nite dot product of monogenic semigroups has recently been dened and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.

On the structure theory of partial automaton semigroups

We study automaton structures, i.e., groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide).

Finite monogenic semigroups and saturated varieties of semigroups

J. M. Howie has shown that the finite monogenic semigroups are absolutely closed. We provide a new, simple and direct proof of Howie’s result. We also show that all the varieties defined by the identities ax = axa, axy = axay and axy = axyay are saturated if and only if they are epimorphically closed.

Automorphism groups of semigroups of upfamilies

A family [Formula: see text] of nonempty subsets of a set [Formula: see text] is called an upfamily if for each set [Formula: see text] any set [Formula: see text] belongs to [Formula: see text]. The extension [Formula: see text] of [Formula: see text] consists of all upfamilies on [Formula: see text]. Any associative binary operation [Formula: see text] can be extended to an associative binary operation [Formula: see text]. In the paper, we study automorphisms of extensions of groups, finite monogenic semigroups, null semigroups, right zero semigroups and left zero semigroups. Also, we describe the automorphism groups of extensions of some semigroups of small cardinalities.

ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

The direct product$\mathbb{N}\times \mathbb{N}$of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for$\mathbb{N}\times S$, where$S$is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if$S$is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of$S$has a relative left or right identity element.

Automorphisms of semigroups of k-linked upfamilies

A family $$ \mathcal{A} $$ of non-empty subsets of a set X is called an upfamily, if, for each set $$ A\in \mathcal{A} $$ ; any set B ⊃ A belongs to $$ \mathcal{A} $$ . An upfamily $$ \mathrm{\mathcal{L}} $$ is called k-linked, if $$ \cap \mathrm{\mathcal{F}}\ne \varnothing $$ for any subfamily $$ \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} $$ of cardinality $$ \left|\mathrm{\mathcal{F}}\right|\le k $$ . The extension Nk(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : Nk(X) × Nk(X) → Nk(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.

Some graph parameters on the strong product of monogenic semigroup graphs

In Das et al. (2013), it
 has been defined a new algebraic graph on monogenic semigroups. Our main scope
 in this study, is to extend this study over the special algebraic graphs to the
 strong product. In detail, we will determinate some important graph parameters
 (diameter, girth, radius, maximum degree, minimum degree, chromatic number,
 clique number and domination number) for the strong product of any two monogenic
 semigroup graphs.

On the dot product of graphs over monogenic semigroups

Now define S a cartesian product of finite times with SMn which is a finite semigroup having elements {0,x,x2,…,xn} of order n. Γ(S) is an undirected graph whose vertices are the nonzero elements of S. It is a new graph type which is the dot product. k be finite positive integer for 0≤{it}t=1k,{jt}t=1k≤n, any two distinct vertices of S(xi1,xi2,…,xik) and (xj1,xj2,…,xjk) are adjacent if and only (xi1,xi2,…,xik)·(xj1,xj2,…,xjk)=0SMn (under the dot product) and it is assumed xit=0SMn if it=0.In this study, the value of diameter, girth, maximum and minimum degrees, domination number, clique and chromatic numbers and in parallel with perfectness of Γ(S) are elucidated.

Concrete algorithms for word problem and subsemigroup problem for semigroups which are disjoint unions of finitely many copies of the free monogenic semigroup

Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has soluble word problem and soluble membership problem. Efficient algorithms are given for both problems.