Find in Library

In recent years, the <i>m</i>-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 <i>m</i>-polar (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula>) structure is a generalization of <i>m</i>-polar fuzziness and cubic structures. The intent of this research is to extend the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> structures to the theory of groups and semigroups. In the present research, we preface the concept of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> groups and probe many of its characteristics. This concept allows the membership grade and non-membership grade sequence to have a set of <i>m</i>-tuple interval-valued real values and a set of <i>m</i>-tuple real values between zero and one. This new notation of group (semigroup) serves as a bridge among <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> structure, classical set and group (semigroup) theory and also shows the effect of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> structure on a group (semigroup) structure. Moreover, we derive some fundamental properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> groups and support them by illustrative examples. Lastly, we vividly construct semigroup and groupoid structures by providing binary operations for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">C</mi><mi>m</mi></msub><mi mathvariant="script">P</mi></mrow></semantics></math></inline-formula> structure and provide some dominant properties of these structures.