The generalized circular intuitionistic fuzzy set and its operations

<abstract><p>The circular intuitionistic fuzzy set (<italic>CIFS</italic>) is an extension of the intuitionistic fuzzy set (<italic>IFS</italic>), where each element is represented as a circle in the <italic>IFS</italic> interpretation triangle (<italic>IFIT</italic>) 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 <italic>IFS</italic>, <italic>CIFS</italic> remains limited to the rigid $ IFIT $ space, where the sum of $ \mathcal{M} $ and $ \mathcal{N} $ cannot exceed one. In contrast, the generalized <italic>IFS</italic> (<italic>GIFS</italic>) allows for a more flexible <italic>IFIT</italic> space based on the relationship between $ \mathcal{M} $ and $ \mathcal{N} $ degrees. To address this limitation, we propose a generalized circular intuitionistic fuzzy set (<italic>GCIFS</italic>) that enables the expansion or narrowing of the <italic>IFIT</italic> area while retaining the characteristics of <italic>CIFS</italic>. Specifically, we utilize the generalized form introduced by Jamkhaneh and Nadarajah. First, we provide the formal definitions of <italic>GCIFS</italic> along with its relations and operations. Second, we introduce arithmetic and geometric means as basic operators for <italic>GCIFS</italic> 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 <italic>GCIFS</italic> and examine their properties. To demonstrate their practical applicability, we provide some examples. In conclusion, we primarily contribute to the expansion of <italic>CIFS</italic> theory by providing generality concerning the relationship of imprecise membership and non-membership degrees.</p></abstract>