Pythagorean and fermatean fuzzy sub-group redefined in context of T ̃-norm and S ̃-conorm

This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG)  in the context of -norm and  -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically Pythagorean Fuzzy Subgroups (PFSG) and FFSG , along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the min and max operations, respectively. However, in this work, we develop a theory that utilizes the -norm for min and the -conorm for max , providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the -norm and -conorm, and discuss important properties associated with them.