Research Of Several Problems Of Many-valued Logic Algebras

Many-valued logic is closely related with some current advanced subjects, such as fuzzy control, artificial intelligence, neurosurgery network and computer science etc. The different fuzzy logic systems are corresponding to the different logic algebras. As far as 1958, the famous logician C. C. Chang had introduced the theory of MV-algebra and succeeded in proving the completeness of Lukasiewicz fuzzy logic system for solving the completeness of Lukasiewicz fuzzy logic system. In 1996, based on the analysis of problems in fuzzy logic and fuzzy reasoning, L^* system and corresponding R0-algebra had been proposed by Professor Wang. With the constantly further research, the completeness of L^* system and corresponding R0-algebra has been proved. And plenty of achievements have been abtained, accelerating the development of fuzzy logic and enriching algebraic contents.The content of this text is divided into four chapters altogether: Chapter one has provided the preliminary knowledge of the lattice theory that will be used behind. Residuated lattice based on continuous t-norms is the important tool to study logical algebra systems in fuzzy logic. For example, BL-algebra, MV-algebra, G-algebra and Goguen algebra are all based on residuated lattice. So the theories of residuated lattices and several kinds of logic algebras are introduced. The relations between several kinds of logic algebras and residuated lattices are discussed in Chapter two. The proper xΛy=x (?) (x→y) in the definition of BL-algebra is so strong that many logic algebras are excluded out. So Sub-BL algebra is proposed, deleting the strong proper xΛy= x(??)(x→y) and retaining the distributive law. By further research, Sub-BL algebra is simplified for that the distributive law can be inferred from the other propers of the definition. In addition, another two equivalent forms of Sub-BL algebra are abtained, revealing the relationship between the Sub-BL algebra and the many-valued algebras. What's more, the simplified definitions of R0-algebra and BR0-algebra based on residuated lattice are achieved.Combining the propers of N-semi-single algebra, the theory of implication algebra and residuated lattice is discussed in N-semi-single algebra in Chapter three. The theory of rings and limited associative algebra is the important branch in algebra. Semi-single algebra holds the important position in limited associative algebra. N-semi-single-algebra can form the system equivalent to FI algebra by the operation "→". And can form the system equivalent to MV algebra the operation "(??)". This paper tries to introduce the operations "→, (??), (??), (??), "in the set G(R) of central idempotent elements in N-semi-single algebra(among the operations,→,⊕and (?) are binary operations. (?)is a unit operation). A dual relation "≤" becoming the partial ordered relation in G(R) is defined. Moreover, (G(R),≤) is proved to be residuated lattice by the corresponding operations. In result, G(R) can become MTL algebra, BL algebra, G algebra, Goguen algebra, BR₀-algebra and R₀-algebra. Through the analysis of the linear ordered BR0-algebra, we proved the completeness of BR₀-algebra according to the proof of the completeness of R₀-algebra and MV-algebra in Chapter four. Thus provides the theory framework for further studying in the corresponding formal deductive systems and fuzzy reasoning. BR₀-algebra is the weak R₀-algebra by removing the proper "(a→b)v ((a→b)→(?) a v b)=1". So the BR₀ unit interval is not unique. This paper provided the proof of the weak completeness of BR₀-algebra.