Groups and Structures of Commutative Semigroups in the Context of Cubic Multi-Polar Structures

Abstract

In recent years, the m-polar fuzziness structure and the cubic structure have piqued the interest of researchers and have been commonly implemented in algebraic structures like groupoids, semigroups, groups, rings and lattices. The cubic m-polar (CmP) structure is a generalization of m-polar fuzziness and cubic structures. The intent of this research is to extend the CmP structures to the theory of groups and semigroups. In the present research, we preface the concept of the CmP groups and probe many of its characteristics. This concept allows the membership grade and non-membership grade sequence to have a set of m-tuple interval-valued real values and a set of m-tuple real values between zero and one. This new notation of group (semigroup) serves as a bridge among CmP structure, classical set and group (semigroup) theory and also shows the effect of the CmP structure on a group (semigroup) structure. Moreover, we derive some fundamental properties of CmP groups and support them by illustrative examples. Lastly, we vividly construct semigroup and groupoid structures by providing binary operations for the CmP structure and provide some dominant properties of these structures.