Homomorphisms on the monoid of fuzzy implications

In this work we propose and study a particular type of lattice and semigroup homomorphisms on the monoid (II, ⊗) of the set of all fuzzy implications proposed in [1]. We show that the subclass of neutral implications which generate homomorphisms of the defined form and the set of such homomorphisms themselves form abelian groups, suggesting that the investigated homomorphisms form the group of inner semigroup homomorphisms. Finally, investigating the images of the studied homomorphisms, we present some natural partitions on I and orderings on these equivalence classes. Our investigations have led us to obtain a group structure on a subset of I. Note that, to the best of the authors' knowledge, this is the first work to present such a rich algebraic structure on the set of all fuzzy implications I.