Homomorphisms on the monoid of fuzzy Implications: a complete characterisation

In [4], we had proposed a novel generating methods of fuzzy implications and investigated algebraic structures on the set of all fuzzy implications, which is denoted by \(\mathbb{I}\). Again in [5], we had defined a particular function g K on the monoid \((\mathbb{I}, \circledast)\) (See Def. 16) and characterised the function K for which g K is a semigroup homomorphism (s.g.h) in two special cases, i.e., K is with trivial range and K(1,y) = y for all y ∈ [0,1](neutrality property). In this work we characterise the nontrivial range non neutral implications K such that g K is an s.g.h. and also present their representations