The decomposition and composition of endomorphism of the fuzzy power set \((F(X), \cup, \cap, c)\) (Q1971940)

The study of homomorphism of various algebraic structures of \(F(X)\), the set of all fuzzy subsets of \(X\), is an extension of the ordinary power set \(P(X)\). For \(I= [0,1]\), \(g: I\to F(X)\), \(f: P(X)\to F(X)\) the transformation \(\sigma\), on \(F(X)\) defined as \(\sigma(A)= \bigvee_{\lambda\in I} g(\lambda) f(A_\lambda)\) is called the composition of \(g\) and \(f\), where \(A_\lambda\) is the \(\lambda\)-cut of \(A\). The strict \(\lambda\)-cut is denoted by \(A_{\underset\bullet\lambda}\). The composition \(\underset\bullet\tau\) is defined as \(\underset\bullet\tau(A)= \bigwedge_{\lambda\in I} g(\lambda) f(A_{\underset\bullet\lambda})\). Introducing a notion called soft homomorphism, sufficient conditions for \(\underset\bullet\tau\) to become a soft homomorphism and sufficient conditions for \(\underset\bullet\tau=\sigma\) are given. Also a theorem which gives necessary and sufficient conditions for a transformation \(\pi\) on \(F(X)\) to become an endomorphism has been proved.