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 $\left\{0e^{i\theta },\left[0,1\right]e^{i\theta },1e^{i\theta }\right\}$. 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.

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 ( $\mathcal{C}\mathcal{C}$) 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 $\mathcal{C}\mathcal{C}$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 ( $\mathcal{C}\mathcal{C}\mathcal{S}$s), weak crossing cubic subalgebras ( $\mathcal{W}\mathcal{C}\mathcal{C}\mathcal{S}$s), and crossing cubic filters ( $\mathcal{C}\mathcal{C}\mathcal{F}$s) in BE-algebras, and we probe several characteristics of these notions. Furthermore, the relationship between $\mathcal{C}\mathcal{C}\mathcal{S}$s, $\mathcal{W}\mathcal{C}\mathcal{C}\mathcal{S}$s, and $\mathcal{C}\mathcal{C}\mathcal{F}$s in BE-algebras is established. After that, the conditions under which $\mathcal{C}\mathcal{C}s$ can be $\mathcal{C}\mathcal{C}\mathcal{S}$ and $\mathcal{C}\mathcal{C}\mathcal{F}$, and the condition under which $\mathcal{W}\mathcal{C}\mathcal{C}\mathcal{S}$ can be $\mathcal{C}\mathcal{C}\mathcal{S}$ are discovered. At last, some characterizations of $\mathcal{C}\mathcal{C}\mathcal{F}$ in BE-algebras are presented.