Continuous Maps in Fuzzy Relations
In this paper we generalize our most recent results that are related to the algorithm that has been developed to automatically derive a fuzzy functional or a fuzzy multivalued dependency from a given set of fuzzy functional and fuzzy multivalued dependencies. Fuzzy dependencies are considered as fuzzy formulas. The first result states that a two-element fuzzy relation instance actively satisfies a fuzzy multivalued dependency if and only if the tuples of the instance are conformant on some known set of attributes with degree of conformance larger than some known constant, and the corresponding fuzzy formula is valid in appropriate interpretations. The second result states that a fuzzy functional or a fuzzy multivalued dependency follows from a set of fuzzy functional and fuzzy multivalued dependencies in two-element fuzzy relation instances if and only if the corresponding fuzzy formula is a logical consequence of the corresponding set of fuzzy formulas. Our earlier research in this direction consisted in an application of some individual fuzzy implication operator, such as Yager, Reichenbach, Kleene-Dienes fuzzy implication operator. The main purpose of this paper is to prove that the aforementioned results remain valid for a wider class of fuzzy implication operators, in particular for the family of f-generated fuzzy implication operators