Research On Construction Of Implication On The Interval Sets

This dissertation comes from"the key scientific and technological project of Henan Province"(No.092102210149)"Research on Flexible Control Model and its System based on Interval Structure"and"the natural science research projects of Education Department of Henan Province"(No.2009A5200 15)"The Algebraic Structure of Interval-valued Fuzzy Logic".In daily life, the problems of the uncertain and incomplete information are difficultly described their exact signification with point-valued, but effectively described with interval-valued. Meanwhile, the information is become effective interval, which reduces its volatility. It ensures scientificity and rationality, and reduces mistakes in decision. So the interval-valued can be widely applied to artificial intelligence and control, neural networks, speech recognition, image recognition, expert systems, biology, sociology and earthquake prediction, etc. The interval-valued reflects the ambiguity in daily reasoning, what's more, it accords with the human thinking habits in the fuzzy information reasoning. Therefore, the interval-valued and theory become one of the hot spots of the research nowadays.In present, interval sets, interval-valued fuzzy sets and interval valued intuitionistic fuzzy sets are researched by scholars, and provide basic theory of the non-classical logic. Interval sets and interval-valued fuzzy sets are an important direction of the study, and have a broad application in fuzzy control, approximate reasoning and other fields. The research of implication is an important part of interval sets and interval-valued fuzzy sets, since the implication only is constructed, further the algebraic system and logic system are researched. However, the interval-valued implication and interval-sets implication are few researched for their construction. According to the implication and basic algebra in the point-valued fuzzy logic, and definition of the inclusion ordering on the interval-valued and interval sets, the in-depth study of implication is extended to the interval. The main innovations are summarized as follows:(1) The Lukasiewicz implication on the point-valued is extended to the interval-valued. So, a new interval-valued Lukasiewicz implication is reconstructed, and its regularities, monotonicities and algebraic properties are discussed.(2) On the interval sets, the new interval implication is redefined, the lattice implication algebras is reconstructed and its properties are discussed. Meanwhile, commutative FI-algebras and MV-algebras also are redefined on the interval sets. Three different algebraic systems are proved to equivalence, which are lattice implication algebras, commutative FI-algebras and MV-algebras.(3) According to the definition of the inclusion ordering on the interval sets, a new generalized interval-sets R-implication is redefined. Regularities and monotonicities of this implication are proved, and the property of adjoint pair of interval-sets intersection and this implication is also proved.(4) A new interval-sets lattice implication is redefined on the interval sets, which is residuated lattice, and is adjoint pair of interval-sets intersection and this implication. Furthermore, some properties of this residuated lattice are discussed.