Invariant characterizations of fuzzy relation information systems under homomorphisms

A homomorphism is an important mathematical tool to study relationships between fuzzy relation information systems. This paper is devoted to investigating reduction in a fuzzy relation information system and its invariant characterizations under homomorphisms. Intersection-reduction, union-reduction, and reduction in a fuzzy relation information system are first proposed. Then, properties of intersection-reduction, union-reduction, and reduction are given. Next, fuzzy relations in a fuzzy relation information system are divided into necessary, relatively necessary, and unnecessary fuzzy relations according to the importance. Finally, some invariant and inverse invariant characterizations of fuzzy relation information systems under consistency and compatible homomorphisms are obtained, respectively. It is worth mentioning that by means of homomorphism, we can get the relatively smaller image system that has the same data structures (i.e., invariant characterizations) as a given original system.