Some solutions of functional equation $I(I(y,x), I(x,y))=I(x,y)$ involving fuzzy implications

TL;DR

This paper investigates solutions to a complex functional equation involving fuzzy implications, providing insights from two perspectives and focusing on main families of fuzzy implications due to the equation's complexity.

Contribution

It introduces two perspectives on the functional equation involving fuzzy implications and explores solutions within main families of fuzzy implications.

Findings

Identified solution structures for the functional equation.

Provided insights into the significance of the equation from two perspectives.

Focused on main families of fuzzy implications due to solution complexity.

Abstract

In this article, a functional equation(IE) involving fuzzy implications has been considered. Two different perspectives of this equation have been provided to realize its significance. As it is very difficult to find the solutions of (IE) in general, the investigation of solutions of (IE) is restricted to main families of fuzzy implications.