In this manuscript, we proposed a novel framework of the q-rung orthopair fuzzy subfield (q-ROFSF) and illustrate that every Pythagorean fuzzy subfield is a q-rung orthopair fuzzy subfield of a certain field. We extend this theory and discuss its diverse basic algebraic characteristics in detail. Furthermore, we prove some fundamental results and establish helpful examples related to them. Moreover, we present the homomorphic images and pre-images of the q-rung orthopair fuzzy subfield (q-ROFSF) under field homomorphism. We provide a novel ideology of a non-standard fuzzy subfield in the extension of the q-rung orthopair fuzzy subfield (q-ROFSF).

In the present paper, we apply the concept of reverse derivation in rings on the concept of d − algebra to obtain the concept called a left-right (resp. right-left) reverse derivations of d−algebra X (briefly, (l, r) resp. (r, l)− reverse derivation of d − algebra ), we will also, define some concepts such as regular map, composition two maps and study the related properties. Moreover, the notions of partial ordered edge d − algebra as well as d − subalgebra and their relation to our current study are obtained. In addition, some illustrative examples and counterexamples are discussed.

In this article, we devise the novel concept of bipolar complex fuzzy (BCF) near rings (BCFNR), to fill a momentous research gap in connection of bipolar complex fuzzy sets with the theory of near rings in the existing literature. We expand the theory of near ring into the structure of BCF set (BCFS), offering a more suitable approach for the representation of algebraic systems with inherent ambiguity, bipolarity, and 2nd dimension information. Further, we introduce the concept of the bipolar complex fuzzy sub-near ring (BCFSNR), bipolar complex fuzzy left ideal (BCFLI), bipolar complex fuzzy right ideal (BCFRI), and bipolar complex fuzzy ideal (BCFI) in the near ring. After that, we prove the related theorem and results of these devised concepts. We also introduce theorems based on the homomorphism theory and Noetherian theory of near rings within the frame of bipolar complex fuzzy near rings. At the end of the manuscript, we reveal the application of BCFNRs in decision-making (DM) and illustrate it through an example.

This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals in INK-algebras and Fermatean neutrosophic sets (FNSs), Fermatean neutrosophic INK-ideals (FNINK-Is), and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-sub algebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of Fermatean neutrosophic INK-ideals and the images of these sub-algebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs.

A relatively new and rapidly expanding area of mathematics research is the study of graphs’ spectral properties. Spectral graph theory plays a very important role in understanding certifiable applications such as cryptography, combinatorial design, and coding theory. Nonassociative algebras, loop groups, and quasigroups are the generalizations of associative algebra. Many studies have focused on the spectral properties of simple graphs connected to associative algebras like finite groups and rings, but the same research direction remains unexplored for loop groups and quasigroups. Eigenvalue analysis, subgraph counting, matrix representation, and the combinatorial approach are key techniques and methods in our work. The main purpose of this paper is to characterize finite quasigroups with the help of relatively prime graphs. Moreover, we investigate the structural and spectral properties of these graphs associated with finite quasigroups in the forms of star graphs, eigenvalues, connectivity, girth, clique, and chromatic number.