Algebraic Aspects of Crossing Cubic BE-Algebras

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.