Algebraic homomorphisms are essential mathematical structures that sustain operations across algebraic systems such as groups, rings, and fields. These mappings not only preserve the validity of algebraic operations but also make it easier to investigate structural similarities and equivalences across distinct algebraic entities. In this article, we establish the group isomorphism under the complex intuitionistic fuzzy set, an extended form of the complex fuzzy set that adds the complex degree of non-membership functions, which plays a significant role in the decision-making process. The complex algebraic structure provides effective tools for understanding complex phenomena. We discuss the more intricate features of homomorphism and isomorphism in the framework of a complex intuitionistic fuzzy set. In addition, we introduce the complex intuitionistic fuzzy normal subgroups. We establish the relationship between two complex intuitionistic fuzzy subgroups and analyze of complex intuitionistic fuzzy isomorphisms among these subgroups, proving the important theorems. Furthermore, we establish examples to explore the concept of complex intuitionistic fuzzy subgroups.

Probabilistic group theory is concerned with the probability of group elements or group subgroups satisfying certain conditions. On the other hand, a polygroup is a generalization of a group and a special case of a hypergroup. This paper generalizes probabilistic group theory to probabilistic polygroup theory. In this regard, we extend the concept of the subgroup commutativity degree of a finite group to the subpolygroup commutativity degree of a finite polygroup $P$. The latter measures the probability of two random subpolygroups $H,K$ of $P$ commuting (i.e., $HK=KH$). First, using the subgroup commutativity table and the subpolygroup commutativity table, we present some results related to the new defined concept for groups and for polygroups. We then consider the special case of a polygroup associated to a group. We study the subpolygroup lattice and relate this to the subgroup lattice of the base group; this includes deriving an explicit formula for the subpolygroup commutativity degree in terms of the subgroup commutativity degree. Finally, we illustrate our results via non-trivial examples by applying the formulas that we prove to the associated polygroups of some well-known groups such as the dihedral group and the symmetric group.

A complex bipolar fuzzy set ( $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$) is an extension of a complex fuzzy set and a $\mathbb{B}\mathbb{F}$ set with a wide range of values. A $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ is differentiated from a $\mathbb{B}\mathbb{F}$ set by the incorporation of negative and positive membership functions to the unit circle in the complex plane, which empowers one to handle the vagueness more effectively. We aim to generalize the notions of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ by proposing a general algebraic structure to dealing with complex bipolar ( $\mathbb{C}\mathbb{B}$) fuzzy data by integrating the idea of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ and subrings. The structure of $\mathbb{C}\mathbb{B}$ fuzzy subrings, such as the $\mathbb{C}\mathbb{B}$ fuzzy ring isomorphism, $\mathbb{C}\mathbb{B}$ fuzzy quotient ring, and $\mathbb{C}\mathbb{B}$ fuzzy ring homomorphism, are examined in this paper. We develop the ( $\delta ,\alpha ;\sigma ,\beta$)-cut of a $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ and explore its algebraic interpretations. Additionally, we describe the $\mathbb{C}\mathbb{B}$ fuzzy support set and demonstrate some significant characteristics associated with this idea. Furthermore, we use the concept of a naturally occurring complex ring homomorphism to describe a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism. Additionally, we prove a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism between the $\mathbb{C}\mathbb{B}$ fuzzy subrings of the ring and the $\mathbb{C}\mathbb{B}$ fuzzy subring of the complex quotient ring. We demonstrate a strong connection among two $\mathbb{C}\mathbb{B}$ fuzzy subrings of complex quotient rings under a specific $\mathbb{C}\mathbb{B}$ fuzzy surjective homomorphism. We construct a complex fuzzy isomorphism between both associated $\mathbb{C}\mathbb{B}$ fuzzy subrings. Furthermore, we introduce three basic results of the $\mathbb{C}\mathbb{B}$ fuzzy isomorphism to explain the relationship between two $\mathbb{C}\mathbb{B}$ fuzzy subrings. Finally, we use a complex bipolar fuzzy subring in decision making.