Journal of Hyperstructures
Volume:11 Issue: 2, Summer and Autumn 2022

The concept of $\Gamma$-semihyperring is a generalization of semiring, semihyperring and a $\Gamma$-semiring. The notion of bi-ideals and minimal bi-ideals in $\Gamma$-semihyperring is introduced with several examples. We also made some ideal theoretic characterization of bi-ideals and minimal bi-ideals in $\Gamma$-semihyperring. Then the notion of bi-simple $\Gamma$-semihyperring is introduced and it is proved that if $R$ be a $\Gamma$-semihyperring without zero. Then $R$ is a bi-simple $\Gamma$-semihyperring if and only if $(k)_{b}=R$, for all $k\in R$.

In this paper we introduce the notion of quasi-hyperideal in multiplicative hypersemirings which is a generalization of onesided hyperideal and study some of its properties and obtain some characterizations of quasi-hyperideal in multiplicative hypersemirings. Also, we introduce the notion of bi-hyperideal in multiplicative hypersemirings. We prove that in a multiplicative hypersemiring every quasi-hyperideal is a bi-hyperideal, but the converse is not true. Lastly, we characterize regular multiplicative hypersemiring with the help of quasi-hyperideal and bi-hyperideal.

Using Vague ideals of near-ring R, authors introduce the concepts of normal vague ideals, complete vague ideals and maximal vague ideals of near-ring R with few properties.

A set P of vertices in a connected graph G is called open monophonic chromatic set if P is both an open monophonic set and a chromatic set. The minimum cardinality among the set of all open monophonic chromatic sets is called open monophonic chromatic number and is denoted by ?om(G). Here properties of open monophonic chromatic number of connected graphs are studied. Open monophonic chromatic number of some standard graphs are identied. For 3 ? m ? n, there is a connected graph G such that ?(G) = m and ?om(G) = n. For 3 ? m ? n, there is a connected graph G such that om(G) = m and ?(G) = ?om(G) = n.Let r,d be two integers such that r < d ? 2r and suppose k ? 2. Then there exists a connected graph G with rad G = r, diam G = d and ?om(G) = k.

The purpose of the present paper is to throw light on the study of vague soft R-subgroup over near-ring R. We have defined vague soft R-subgroup over a near-ring. By using quotient near-ring we have defined vague soft quotient near-ring. Also, we have investigated operations of vague soft R-subgroup over a near- ring and vague soft quotient near-ring.

In this work, we give a fuzzy analogy of cone symmetric spaces that we call fuzzy cone symmetric spaces. Since these structures are obtained by omitting the triangle inequality in fuzzy cone metric spaces, there are topological degenerations. After mentioning these degenerations, we investigate the relationship between cone (sym)metric and fuzzy cone (sym)metric spaces.

In this paper, we introduce the notion of hyper JK- algebras and investigate these algebras properties. Moreover, we present relationships between hyper JK-algebras and pseudo hyper BCK-algebras and hyper pseudo MV-algebras under some condi- tions.

In this paper, a common fixed point theorem for four commutative mappings in the setting of digital metric space is proved with a supportive example. Also, we established some common fixed point theorems for weakly compatible mappings that satisfy certain contractive conditions in digital metric space.

Intuitionistic fuzzy sets (IFSs) introduced by Atanassov are generalisations of fuzzy sets which are powerful tools in dealing with vagueness. In this paper, concept of convex (concave) IFSs and its characteristics using cut sets of IFSs were studied. In par- ticular, we introduced affine intuitionistic fuzzy sets and investigate some of its characteristics.

The object of the present paper is to study the Riemannian solitons on para Sasakian manifolds admitting E?R=0, E?P=0, E?E=0 , E?P^{?}=0, E?M=0, E?W_{i}=0, E?W_{i}^{?}=0, R?R=0, R?P=0 , R?E=0, R?P^{?}=0, R?M=0, R?W_{i}=0, R?W_{i}^{?}=0, R?K=0, R?C=0, E?C=0 and E?K=0, ( for all i=1,2,....9).

The object of the present paper is to study Lorentzian para-Sasakian manifold on a pseudo slant submanifold and using some properties like warped product on manifolds, totally geodesic foliation, integrability on the properties of nearly Lorentzian para- Sasakian manifold we find some results.

It is well known that the exponential Diophantine equation $2^{x}+ 1=z^{2}$ has the unique solution $x=3$ and $z=3$ in non-negative integers, which is closely related to the Catlan's conjecture. In this paper,
we show that for $m \in \mathbb{N}, m > 1$, the exponential Diophantine equation $2^{x}+m^{2y}=z^{2}$ admits a solution in positive integers $(x, y,z)$ if and only if $m=2^{\alpha}M_{n}, \alpha \geq 0$ for some Mersenne number $M_{n}$. When $m=2^{\alpha}M_{n}, \alpha \geq 0$, the unique solution is $(x,y,z)=(2+n+2 \alpha,1, 2^{\alpha}(2^{n}+1))$.
Finally, we conclude with certain examples and non-examples alike! The novelty of the paper is that we mainly use elementary methods to solve a particular class of exponential Diophantine equations.