The Research About Some Logic Algebras And Systems

The theories and technologies of Fuzzy systems control haveachieved the world-famous success. Then Fuzzy inference and Fuzzy logic as thekernel of the theories of Fuzzy system control have been concerned more and more.In the development process of Fuzzy inference, many propositional logic systemshave appear. Among numerous propositional logic systems, Lukasiewicz,Godel,Product and L^* have obviously merits, which are exist triangle norm * in [0, 1] withthe semantic implication operator→construct adjoint pairs. The preceding threesystems correspondence triangle norms are continues. Petre Hajek have proposedBL algebra based on the three continues triangle norms and construct Basic Logicsystems correspondingly. Afterwards, Professor Hong-Bo WU proposed BL^* systemthat are directed at Lukasiewicz system and L^* system which completeness are betterresolved. Both Basic Logic system and BL^* system are based on residuated latticesand Fuzzy Implication algebras. Then, what differences and connection they make? Whether can be united? The main point of the context is it.The context started from the properties of the logic algebra which based onResiduated lattic, studied the relationships between all kind of logic algebras andits corresponding logic systems. The main results of the text are: First, it domore research on the properties of Residuated lattices, and proposed the concept ofPrelinearityresiduated lattice, then proved the completeness of Prelinearity residu-ated lattice that correspond to the Orderd residuate lattice. Second, the contextconstructed the PL^* system which based on Prelinearity residuated lattice, andproved it completeness. Third, proved Prelinearity residuated lattice is the basisof BL-algebra and BR0-algebra, and PL^* system is the basis of BL-system andBR₀-system. Then Prelinearity residuated lattice is the common basis of MV-algebra,R₀-algebra,G-algebra and H-algebra, and PL^* system is the commonbasis of Lukasiewicz system,G(o|¨)del system,Product system and L^* system.Fourth, gave some simplified definition of MV-algebra and R₀-algebra, and pro- posed the concept of Weak Lattic Implication-Algebra, and proved the equivalenceof Weak Lattic Implication-Algebra and BR₀-algebra.The construction and the main contents of this paper are as follows:Chapter 1. Preliminary knowledge. In this chapter, we make a depiction forbasic knowledge of and basic properties of Residuated-lattice,Fuzzy implication-algebra. BL-algebra and BR₀-algebra which would be used in the following chap-ters. And the relationship between Residuated-lattice and Fuzzy implication-algebrahave been studied.Chapter 2. The concept of Prelinearity residuated lattice have been pro-posed, and its completeness which correspond to ordered residuated lattice havebeen proved.Chapter 3. Erect the PL^* system based on Prelinearity residuated lattice, andproved it's completeness.Chapter 4. Proved that Prelinearity residuated lattice is the basis of BL-algebra and BR0-algebra, and PL^* system is the basis of BL-system and BR₀-system.Chapter 5. Gives some simplifier definition of MV-algebra and R₀-algebra,and proposed the concept of Weak Lattic Implication-Algebra, and proved the equiv-alence of Weak Lattic Implication-Algebra and BR₀-algebra.