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

Abstract

<abstract><p>The circular intuitionistic fuzzy set (<italic>CIFS</italic>) extends the concept of <italic>IFS</italic>, representing each set element with a circular area on the <italic>IFS</italic> interpretation triangle (<italic>IFIT</italic>). Each element in <italic>CIFS</italic> 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 <italic>CIFS</italic>, 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 <italic>CIFS</italic>, focusing on operations for membership and non-membership degrees as well as radius domains. First, we proposed new radius operations on <italic>CIFS</italic> with a domain $ [0, \psi] $, where $ \psi \in [1, \sqrt{2}] $, called a radius algebraic product (<italic>RAP</italic>) and radius algebraic sum (<italic>RAS</italic>). Second, we developed basic operators for generalized union and intersection operations on <italic>CIFS</italic> 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 <italic>CIFS</italic>, providing valuable insights into its potential applications, particularly in decision-making theory.</p></abstract>