A certain algebraic structure of bipolar neutrosophic subgroups

The notion of a bipolar neutrosophic set (BNS) was created as an expansion of a neutrosophic set when everysingle-value neutrosophic membership function has two poles (a positive pole and a negative pole). The BNS environment isan innovative tool for addressing ambiguous situations in several decision-making problems, especially in situations wherethe human mind perceives two patterns of thinking: positive and negative. In this study, we apply this idea in an algebraicenvironment when we initiate the novel concept of bipolar neutrosophic subgroups and prove that every BN subgroupgenerates two BN subgroups. We explain several ideas based on this environment, such as the level set, support, kernelfor BNS, BN characteristic function, and BN-point. Then, we illuminate the BN-subgroup, BN-normal subgroup, BSVNconjugate,normalizer for BN-subgroup, BN-abelian subgroup, and BN -factor group. Furthermore, we present the linkedtheorems and examples and prove these theorems. Moreover, we discussed the image and pre-image of BN-subgroups underhomomorphism and proved the related theorems.Continuing our application of algebraic concepts to BNS, we will present anew mathematical framework for the concept of BN-matrecs and some related mathematical properties. In addition, we willprovide a practical application of these tools in dealing with a decision-making (DM) problem. Finally, in the conclusionsection, we recommended that these results be a qualitative addition to the fuzzy algebraic environment and that they openup research doors for other future research work that takes advantage of the bipolar environment.