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A. K. Bhuniya

Publications and source records attributed to A. K. Bhuniya.

13 recordsLinked to original sources
arXiv2023

On the Jacobson radical and semisimplicity of a semiring

Based on the minimal and simple representations, we introduce two Jacobson-type Hoehnke radicals, m-radical and s-radical, of a semiring $S$. Every minimal (simple) $S$-semimodule is a quotient of $S$ by a regular right congruence (maximal) $μ$ on $S$ such that $[0]_μ$ is a maximal $μ$-saturated right ideal in $S$. Thus the m(s)-radical becomes an intersection of some regular congruences. Finally, every semisimple semiring is characterized as a subdirect product of primitive semirings; and every s-primitive semiring is represented as a 1-fold transitive subsemiring of the semiring of all endomorphisms on a semimodule over a division semiring.

math.RA↗
arXiv2018

Uniqueness of primary decompositions in Laskerian le-modules

Here we introduce and characterize a new class of le-modules $_{R}M$ where $R$ is a commutative ring with $1$ and $(M,+,\leqslant,e)$ is a lattice ordered semigroup with the greatest element $e$. Several notions are defined and uniqueness theorems for primary decompositions of a submodule element in a Laskerian le-module are established.

math.RA↗
arXiv2018

On the prime spectrum of an le-module

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule elements of $M$. Thus we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec(M) is connected if and only if $R/Ann(M)$ contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring $R/Ann(M)$ induces a base of quasi-compact open sets for the Zariski-topology on Spec(M). Every irreducible closed subset of Spec(M) has a generic point. Besides, we prove a number of different equivalent characterizations for Spec(M) to be spectral.

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arXiv2017

On the idempotent semirings such that $\mathcal{D}^\bullet$ is the least distributive lattice congruence

Here we describe the least distributive lattice congruence $η$ on an idempotent semiring in general and characterize the varieties $D^\bullet, L^\bullet$ and $R^\bullet$ of all idempotent semirings such that $η=\mathcal{D}^\bullet, \mathcal{L}^\bullet$ and $\mathcal{R}^\bullet$, respectively. If $S \in D^\bullet [L^\bullet, R^\bullet]$, then the multiplicative reduct $(S, \cdot)$ is a [left, right] normal band. Every semiring $S \in D^\bullet$ is a spined product of a semiring in $L^\bullet$ and a semiring in $R^\bullet$ with respect to a distributive lattice.

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arXiv2017

On band orthorings

A semiring $S$ which is a union of rings is called completely regular, if moreover, it is orthodox then $S$ is called an orthoring. Here we study the orthorings $S$ such that $E^+(S)$ is a band semiring. Every band semiring is a spined product of a left band semiring and a right band semiring with respect to a distributive lattice. A similar spined product decomposition for the band orthorings have been proved. The interval $[\mathbf{Ri}, \mathbf{BOR}]$ is lattice isomorphic to the lattice $\mathcal{L}(\mathbf{BI})$ of all varieties of band semirings, where $\mathbf{Ri}$ and $\mathbf{BOR}$ are the varieties of all rings and band orthorings, respectively.

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arXiv2017

On the spectrum of linear dependence graph of finite dimensional vector spaces

In this paper, we introduce a graph structure called linear dependence graph of a finite dimensional vector space over a finite field. Some basic properties of the graph like connectedness, completeness, planarity, clique number, chromatic number etc. have been studied. It is shown that two vector spaces are isomorphic if and only if their corresponding linear dependence graphs are isomorphic. Also adjacency spectrum, Laplacian spectrum and distance spectrum of the linear dependence graph have been studied.

math.CO↗
arXiv2016

On the power graph of the direct product of two groups

The power graph $P(G)$ of a finite group $G$ is the graph with vertex set $G$ and two distinct vertices are adjacent if either of them is a power of the other. Here we show that the power graph $P(G_1 \times G_2)$ of the direct product of two groups $G_1$ and $G_2$ is not isomorphic to either of the direct, cartesian and normal product of their power graphs $P(G_1)$ and $P(G_2)$. A new product of graphs, namely generalized product, has been introduced and we prove that the power graph $P(G_1 \times G_2)$ is isomorphic to a generalized product of $P(G_1)$ and $P(G_2)$.

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arXiv2016

On some properties of enhanced power graph

Given a group $G$, the enhanced power graph of $G$ denoted by $\mathcal{G}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x, y$ are edge connected in $\mathcal{G}_e(G)$ if there exists $z\in G $ such that $x=z^m$ and $ y=z^n $, for some $m, n\in \mathbb{N}$. In this article, we characterize the enhanced power graph $\mathcal{G}_e(G)$ of $G$. The graph $\mathcal{G}_e(G)$ is complete if and only if $G$ is cyclic, and $\mathcal{G}_e(G)$ is Eulerian if and only if $|G|$ is odd. We classify all abelian groups and also all non-abelian $p-$groups $G$ for which $\mathcal{G}_e(G)$ satisfies the cone property.

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arXiv2016

Normal Subgroup Based Power Graph of a finite Group

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $Γ_H(G)$ whose vertex set $V(Γ_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \in \mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $Γ_H(G)$ and the power graph $Γ(\frac{G}{H})$. We show that $Γ_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\in \mathbb{N}$. $Γ_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\frac{G}{H}\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. Also $Γ_H(G)$ is Eulerian if and only if $|G|\equiv |H|$ mod$ 2$.

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arXiv2015

On some characterizations of strong power graphs of finite groups

Let $ G $ be a finite group of order $ n$. The strong power graph $\mathcal{P}_s(G) $ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2 < n$. In this article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.

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