Complex fuzzy sets (CFS) generalize traditional fuzzy sets (FS) since the membership functions of CFS reduces to the membership functions of FS. FS values are always at [0, 1], unlike CFS which has values in the unit disk of C. This paper merges notion and concept in group theory and presents the notion of a complex fuzzy subgroup of a group. This proposed idea represents a more general and better optional mathematical tool as one of the approaches in the fuzzy group. However, this research defines the notion of complex fuzzy subgroupiod, complex fuzzy normal subgroup, and complex fuzzy left(right) ideal. Therefore, the lattice, homomorphic preimage, and image of complex fuzzy subgroupiod and ideal are introduced and studied its properties. Finally, complex fuzzy subgroups and their properties are presented and investigated

Modern technology makes it easier to store datasets, but extracting and isolating useful information with its full meaning from this data is crucial and hard. Recently, several algorithms for clustering data have used complex fuzzy sets (CFS) to improve clustering performance. Thus, adding a second dimension (phase term) to the range of membership avoids the problem of losing the full meaning of complicated information during the decision-making process. In this research, the notion of the complex shadowed set (CSHS) was introduced and considered as an example of the three region approximations method simplifying processing with the support of CFS and improving the representation of results attained within. This notion can be founded by extending the shadowed set codomain from $ \left\{0, \left[0, 1\right], 1\right\} $ into $ \left\{0{e}^{i\theta }, \left[0, 1\right]{e}^{i\theta }, 1{e}^{i\theta }\right\} $. The significance of CSHS was illustrated by giving an example. Additionally, some properties of the CSHS were examined. The basic CSHS operations, complement, union, and intersection were investigated with their properties. Finally, an application in decision-making was illuminated to support the present notion.

Correctly determining a company's market worth during an entire year or a certain period presents a difficulty to decision-makers. In the case of the merger of companies, the need performs heavier when both the companies' owners are attracted to establishing a fair price at the optimal time to merge. The effectiveness of representing, connecting and manipulating both uncertainty and periodicity information becomes highly required. Hence, study and nhance some properties and conditions of the algebraic structure of complex hesitant fuzzy graphs. Therefore, the degree of composition between two complex hesitant fuzzy graphs is proposed. Also, the formal definitions of union, joint and complement are presented to be covered in the realm of complex hesitant fuzzy graphs. A real-life application is illustrated to show the relation between vertices and edges in the form of complex hesitant fuzzy graphs.

Recently, interval-valued fuzziness and negative structures have gained popularity among researchers and have been widely used in algebraic structures such as semigroups, rings, and lattices. The crossing cubic structure $(\mathcal{CC})$ is an expansion of a bipolar fuzziness structure and a parallel circuit between interval-valued fuzziness and negative structures. The main objective of this study is to apply the idea of $\mathcal{CC}\ \mathrm{s}$ to BE-algebras. In the present research, we modify and extend the notions of fuzziness algebraic substructures, namely subalgebras, weak subalgebras and filters of BE-algebras to introduce the notions of crossing cubic subalgebras $(\mathcal{CCS}\ \mathrm{s})$, weak crossing cubic subalgebras $(\mathcal{WCCS}\ \mathrm{s})$, and crossing cubic filters $(\mathcal{CCF}\ \mathrm{s})$ in BE-algebras, and we probe several characteristics of these notions. Furthermore, the relationship between $\mathcal{CCS}\ \mathrm{s},\mathcal{WCCS}\ \mathrm{s}$, and $\mathcal{CCF}\ \mathrm{s}$ in BE-algebras is established. After that, the conditions under which $\mathcal{CC}s$ can be $\mathcal{CCS}$ and $\mathcal{CCF}$, and the condition under which $\mathcal{WCCS}$ can be $\mathcal{CCS}$ are discovered. At last, some characterizations of $\mathcal{CCF}$ in BE-algebras are presented.

Accurately assessing academic performance is a persistent challenge due to the uncertain and imprecise nature of evaluative information. While Hesitant Fuzzy Sets (HFS) provide a useful tool for handling uncertainty, existing hesitant fuzzy relation (HFR) structures often fail to preserve original information, leading to incomplete evaluations. The main objective of this research is to develop a new HFR structure that preserves original data more effectively, enabling more accurate, reasonable, and complete assessments. We generalize the Cartesian product and hesitant fuzzy relations by introducing a hesitant fuzzy Cartesian product (HFCP), where each pair ( x , y ) is associated with hesitant fuzzy elements. This extends classical and intuitionistic fuzzy models using a lattice-based mathematical framework. The analytical approach relies on set theory and lattice structures. An illustrative example using simulated academic data demonstrates the improved representation and accuracy of lecturer–student evaluations. Key findings highlight that the new structure reduces information loss and increases the reliability of performance assessment. This framework supports better-informed decision-making in academic management. We recommend incorporating hesitant fuzzy-based evaluation models in institutional appraisal systems to enhance fairness, clarity, and data completeness in academic performance reviews.