The Isomorphism and Homomorphism of Fuzzy Sets

In this paper, the concepts of the isomorphism and homomorphism of fuzzy sets are given. The sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets, and some properties of the new structural definition are discussed. The essence of fuzzy sets is point out and the operational properties of fuzzy sets based on isomorphism and homomorphism of fuzzy sets are discussed. Fortunately, we studied the isomorphism and homomorphism of fuzzy relations, have given the concept of isomorphic classification of fuzzy similarity relation, proved that if two fuzzy similarity relation are isomorphic, then they must be classified isomorphic. And, we obtain a conclusions: any fuzzy similarity relation is classified isomorphic to its transitive closure, which ensure we may get the classified structure of X from the original fuzzy similar matrix in theory.