Studies of R-implication based on the left continuous t-norm

The properties and applications of R-implication based on the left continuous t-norm are studied in this paper. Firstly, fourteen properties which are very important to the studies of implication operators are listed. Some interdependencies among the properties are also discussed. As a result, we obtain three mutually independent properties that can infer all the fourteen properties. Then we discuss that whether the R-implication based on the left continuous t-norm satisfy the fourteen properties. Finally the applications of the R-implication operators based on the left continuous t-norm are discussed, the triple-I method based on them are given, Moreover, the properties of triple-I algorithms based on the R-implication induced by the left continuous t-norm for FMP and FMT are given. we also compare the errors of three fuzzy systems when they use the same fuzzy reasoning rules.