On The Monotonicity Of Fuzzy Systems Based Upon Genuine Many-valued Implications-SISO Cases

Recently, Yong-Ming Li proposed fuzzy systems based upon genuine many-valued implica-tions for SISO cases[11]. From the opinion of application, it is expected that the fuzzy systemshave monotonicity if both the antecedent and consequent parts of fuzzy rules are monotone.Hence we discuss the monotonicity of this fuzzy systems in this paper. Specifically speaking, threetopics are discussedμone is the monotonicity of the fuzzy systems based upon R-implications,another is the monotonicity of the fuzzy systems based upon S-implications, the last is themonotonicity of the fuzzy systems based upon QL-implications.In chapter1. The fuzzy systems based upon genuine many-valued implications-SISO Casesand its rules are reviewed. The definition of fuzzy sets is introduced. In particular, the definitionof normal base set are given. Finally, genuine many-valued implications (i.e., R-implications,S-implications, QL-implications) are recalled.In chapter2. We explore the monotonicity of the fuzzy systems based upon R-implications.First, we point out that the fuzzy systems based upon R-implications need not have the mono-tonicity in general. Secondly, we give out the sufcient conditions that the fuzzy systems basedupon R-implications have monotonicity.In chapter3. We explore the monotonicity of the fuzzy systems based upon S-implications.First, we point out that the fuzzy systems based upon S-implications need not have the mono-tonicity in general. Secondly, we give out the sufcient conditions that the fuzzy systemsbased upon Reichenbach implications aâ†'b=1-a+ab and Kleene-Dienes implicationsaâ†'b=(1-a)∨b have monotonicity.In chapter4. We explore the monotonicity of the fuzzy systems based upon QL-implications.First, we point out that the fuzzy systems based upon QL-implications need not have the mono-tonicity in general. Secondly, we give out the sufcient conditions that the fuzzy systems basedupon QL-implications I(a, b)=1a+a2b and Zadeh implications I(a, b)=(1a)∨(a∧b)have monotonicity.It is pointed out that our proofs are diferent with, strictly speaking, more complex thanH. Seki’s proofs because there does not exist the uniform formula of output in Li’s system whileH. Seki’s proofs have them.