The generalized circular intuitionistic fuzzy set and its operations

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.