Incorporating complex N-fuzzy set with classical semigroups and application in real life industrial systems

In recent years, complex fuzzy sets and $N$-fuzzy sets have gained substantial attention within the research community, leading to their extensive utilization in different algebraic systems including, groups, rings, and modules. This research work presents a pioneering mathematical framework, the complex $N$-fuzzy set, which combines the complex fuzziness of parameters with the negative fuzziness of data. Complex $N$-fuzzy sets represent a recent generalization of complex fuzzy sets and $N$-fuzzy sets in which the membership degree of each element is allowed to take values in a complex negative domain of the form $\left[-1,0\right]+[-1,0]i$. The real part represents a negative, counter-supportive or inhibitive degree of membership, while the imaginary part models an independent direction of orthogonal uncertainty, conflicting evidence, or dual hesitation. Based on this new structure, we introduce and examine the notions of complex $N$-fuzzy characteristic functions, level ( $\alpha ,\beta$)-cut, and the product of two complex $N$-fuzzy sets. Also, we present some fundamental operations of complex $N$-fuzzy sets and certain examples of them. In addition, this paper presents the novel mathematical framework of complex $N$-fuzzy semigroups, an algebraic structure that arises by superimposing complex $N$-fuzzy sets on crisp semigroup theory. Here, we present the notions of complex $N$-fuzzy sub-semigroups, complex $N$-fuzzy left (right) ideals, and complex $N$-fuzzy ideals of semigroups. We study certain theorems and their corresponding proofs to underpin these foundational concepts. Furthermore, we investigate some characterizations of these concepts using level ( $\alpha ,\beta$)-cut, and the product of two complex $N$-fuzzy sets. Lastly, we use complex $N$-fuzzy semigroup structure in real life industrial systems.