A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of $(\alpha ,\beta )$-complex fuzzy sets and then define $\left(\alpha ,\beta \right)$-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an $(\alpha ,\beta )$-complex fuzzy subgroup and define $(\alpha ,\beta )$-complex fuzzy normal subgroups of given group. We extend this ideology to define $(\alpha ,\beta )$-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that $(\alpha ,\beta )$-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of $(\alpha ,\beta )$-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the $(\alpha ,\beta )$-complex fuzzy subgroup of the classical quotient group and show that the set of all $(\alpha ,\beta )$-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of $\left(\alpha ,\beta \right)$-complex fuzzy subgroups and investigate the $(\alpha ,\beta )$-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

In this paper, we introduce the idea of \(Q\)-complex fuzzy sub-ring (\(Q\)-CFSR) and discuss its various algebraic aspects. We prove that every \(Q\)-CFSR generates two \(Q\)-fuzzy sub-rings (\(Q\)-FSRs). We also present the concept of level subsets of \(Q\)-CFSR and show that level subset of \(Q\)-CFSR form sub-ring. Furthermore, we extend this idea to define the notion of the direct product of two \(Q\)-CFSR Moreover, we investigate the homomorphic image and inverse image of \(Q\)-CFSR.

In this work, we incorporate modular arithmetic and discuss a special class of graphs based on power functions in a given modulus, called power digraphs. In power digraphs, the study of cyclic structures and enumeration of components is a difficult task. In this manuscript, we attempt to solve the problem for $ p $th power congruences over different classes of residues, where $ p $ is an odd prime. For any positive integer $ m $, we build a digraph $ G(p, m) $ whose vertex set is $ \mathbb{Z}_{m} = \{0, 1, 2, 3, ..., m-1\} $ and there will be a directed edge from vertices $ u\in \mathbb{Z}_{m} $ to $ v\in \mathbb{Z}_{m} $ if and only if $ u^{p}\equiv v\; (\text{mod} \; m) $. We study the structures of $ G(p, m) $. For the classes of numbers $ 2^{r} $ and $ p^{r} $ where $ r\in \mathbb{Z^{+}} $, we classify cyclic vertices and enumerate components of $ G(p, m) $. Additionally, we investigate two induced subdigraphs of $ G(p, m) $ whose vertices are coprime to $ m $ and not coprime to $ m $, respectively. Finally, we characterize regularity and semiregularity of $ G(p, m) $ and establish some necessary conditions for cyclicity of $ G(p, m) $.

In this study, we define the concept of an $\omega$-fuzzy set $\omega$-fuzzy subring and show that the intersection of two $\omega$-fuzzy subrings is also an $\omega$-fuzzy subring of a given ring. Moreover, we give the notion of an $\omega$-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an $\omega$-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an $\omega$-fuzzy quotient subring. We also define the idea of a support set of an $\omega$-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe $\omega$-fuzzy homomorphism and $\omega$-fuzzy isomorphism. We establish an $\omega$-fuzzy homomorphism between an $\omega$-fuzzy subring of the quotient ring and an $\omega$-fuzzy subring of this ring. We constitute a significant relationship between two $\omega$-fuzzy subrings of quotient rings under the given $\omega$-fuzzy surjective homomorphism and prove some more fundamental theorems of $\omega$-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of $\omega$-fuzzy isomorphism.

In this study, we investigate two graphs, one of which has units of a ring as vertices (or nodes) and an edge will be built between two vertices and if and only if . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring of the form , where are the units of . Two vertices and are adjacent to each other if and only if . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer in terms of complete graphs.