| Fuzzy logic, as an important branch of non-classical mathematical logics, is the basis of inference mechanism of many fields such as artificial intelligence, information science, etc.. Zadeh distinguished fuzzy logic into broad and narrow senses. Fuzzy logic in narrow sense, FLn, can be considered as a special kind of symbolic logic which aims at a formalization of approximate reasoning, and the research of FLn is strongly connected with the corresponding algebraic structures. In FLn, the standard set of truth-values is the unit interval [0,1] and the truth functions of connective make it to be an algebra of truth degrees. Each FLn has its own algebra (or class of algebras) of truth degrees with the domain [0,1], called the standard algebra(s)of the logic.In 1996, trying to build the strict logical foundation of fuzzy reasoning, Professor Wang Guojuu constructed the formal deductive system L* of fuzzy prepositional calculi and proposed a new fuzzy implication operator, called R0 implication. Then, based on the theory of generalized tautologies, he proposed a kind of new method for fuzzy reasoning, called triple I method, which improved Zadeh's CRI method and provided sernantically a reasonable logic foundation for fuzzy reasoning. R0 implication as the residua of nilpotent minimum t-norm has many significant properties and is an ideal implication operator for fuzzy logic. On the other hand, the logic MTL has been recently introduced by Esteva and Godo as the logic of left continuous t-norms and their residua. Moreover, an extension of MTL, called NM logic has also been obtained to cope with the nilpotent minimum t-norm and its residua. In fact, the system L* and the system NM are equivalent, and the related algebras have the same kind of algebraic structures.This work continues to focus on the research of L* and R0-algebras. The theory of generalized tautologies in .R0-interval W is studied exhaustively and an extension of the system L* is proposed and the corresponding algebra is investigated as well. This paper is divided into three chapters.The first chapter, for the convenience of the research, is devoted to list theknowledge which is necessary in the subsequent chapters. The axioms and certain important theorems of the system L*, MTL logic, WNM logic and NM logic as well as basic definitions and preliminary properties of the corresponding algebras are introduced.The second chapter studies the theory of generalized tautologies w.r.t. the infinite subalgebras of the .R0-interval. (1) The concept and some examples of the regular R0-algebra axe introduced and the characterization theorem of regular R0-algebras is proved. Then we confine ourselves mainly to the theory of generalized tautologies w.r.t. the regular .R0-algebras and show that the generalized tautologies w.r.t. W and W coincide. (2) In the infinite subalgebra E of W, an evaluation representation theorem of A in F(S) is proved firstly, then based on the difference of condensation points of E, a classification of infinite subalgebras of W is presented. This part will play an important role in the research of the theory of generalized tautologies w.r.t. infinite subalgebras of W. (3) The theory of generalized tautologies w.r.t. each kind of infinite subalgebras of W is investigated. It is proved that there are at most four generalized tautologies w.r.t. the normal infinite subalgebras; there exist n generalized tautologies w.r.t. the first kind of infinite subalgebras for each n 3; there are countably many generalized tantologies w.r.t. the second kind of infinite subalgebras.The third chapter proposes an extension of L* and its algebraic structure has been studied. (1) To interpret the linguistic hedge "quite" in natural language, it is enriched the language of ? by adding a new unary connective and obtained the system L*. In system L*, the generalized deduction theorem and other preliminary theorems are proved. (2) The concept of R0-algebra and certain examples are given. By means of the tool of filters, the completeness of...
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