Extension operators of circular intuitionistic fuzzy sets with triangular norms and conorms: Exploring a domain radius

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.