The circular intuitionistic fuzzy set (CIFS) extends the concept of IFS, representing each set element with a circular area on the IFS interpretation triangle (IFIT). Each element in CIFS is characterized not only by membership and non-membership degrees but also by a radius, indicating the imprecise areas of these degrees. While some basic operations have been defined for CIFS, not all have been thoroughly explored and generalized. The radius domain has been extended from $[0,1]$ to $[0,\sqrt{2}]$. However, the operations on the radius domain are limited to $min$ and $max$. We aimed to address these limitations and further explore the theory of CIFS, focusing on operations for membership and non-membership degrees as well as radius domains. First, we proposed new radius operations on CIFS with a domain $[0,\psi ]$, where $\psi \in [1,\sqrt{2}]$, called a radius algebraic product (RAP) and radius algebraic sum (RAS). Second, we developed basic operators for generalized union and intersection operations on CIFS based on triangular norms and conorms, investigating their algebraic properties. Finally, we explored negation and modal operators based on proposed radius conditions and examined their characteristics. This research contributes to a more explicit understanding of the properties and capabilities of CIFS, providing valuable insights into its potential applications, particularly in decision-making theory.

The circular intuitionistic fuzzy set (CIFS) is an extension of the intuitionistic fuzzy set (IFS), where each element is represented as a circle in the IFS interpretation triangle (IFIT) instead of a point. The center of the circle corresponds to the coordinate formed by membership ( $\mathcal{M}$) and non-membership ( $\mathcal{N}$) degrees, while the radius, $r$, represents the imprecise area around the coordinate. However, despite enhancing the representation of IFS, CIFS remains limited to the rigid $IFIT$ space, where the sum of $\mathcal{M}$ and $\mathcal{N}$ cannot exceed one. In contrast, the generalized IFS (GIFS) allows for a more flexible IFIT space based on the relationship between $\mathcal{M}$ and $\mathcal{N}$ degrees. To address this limitation, we propose a generalized circular intuitionistic fuzzy set (GCIFS) that enables the expansion or narrowing of the IFIT area while retaining the characteristics of CIFS. Specifically, we utilize the generalized form introduced by Jamkhaneh and Nadarajah. First, we provide the formal definitions of GCIFS along with its relations and operations. Second, we introduce arithmetic and geometric means as basic operators for GCIFS and then extend them to the generalized arithmetic and geometric means. We thoroughly analyze their properties, including idempotency, inclusion, commutativity, absorption and distributivity. Third, we define and investigate some modal operators of GCIFS and examine their properties. To demonstrate their practical applicability, we provide some examples. In conclusion, we primarily contribute to the expansion of CIFS theory by providing generality concerning the relationship of imprecise membership and non-membership degrees.