Logic Algebra

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 corresponding with the Lukasiewicz logic system. Afterwards, in order to supply every probable logic system for fuzzy reasoning, many scholars introduced every different algebra system. For instance, Professor Wu Wang-ming introduced fuzzy implication algebra. Professor Xu Yang introduced lattice implication algebra. In 1997, based on the analysis of problems in fuzzy logic and fuzzy reasoning ,L^* system and corresponding R₀ algebra had been proposed by Professor Wang Guo-jun. With the constantly further research, the completeness of L^* system and corresponding R₀ algebra had been proved. And plenty of achievements had been abtained. In the research of logic reasoning system and logic algebra system, both filter and ideal are important concepts. Many experts and scholars had made certain research in this aspect. This paper will make further research about filter, ideal and subalgebra in many-valued logic algebra based on the achievements that had been obtained.The content of this text is divided into five chapters altogether. Chapter one has provided the preliminary knowledge of the lattice theory that will be used behind. Several kinds of logic algebras and the properties of these algebras are given.In chapter two, first of all,according to the definition of adjointing operator(*,�'), the concept of ^*filter is introduced by the binary ^* operation in MV algebra. It is proved that ^* filter is equal to common filter. And some fundamental properties of ^*filter are discussed. Second, the concept of^*filter is defined in Lindenbaum algebra F(S)/- about Lukasiewicz system. With the corresponding knowledge of lattice, the properties of D(Γ) are studied. The main results are : (1)the ^*filter in Luk-Lindenbaum algebra F(S)/ - is the same style as < D(Γ) >; (2)the ^*filter coincideswith MP filter,both of them are common filters,but the opposite conclusion does not hoid; (3) the maximal ^*filter in F(S)/ - is equal to the maximal filter which in common sense.In chapter three,combining the properties of fuzzy sets and ideals in logic algebra,the concepts of fuzzy ideal and fuzzy prime ideal in BR₀-algebra are introduced. Some properties of fuzzy ideal and fuzzy prime ideal are discussed. The sufficient and necessary conditions for fuzzy sets to be fuzzy ideals are given. It is proved that fuzzy ideal and fuzzy prime ideal are invariant under BR₀-algebra isomorphism.In chapter four,first of all,combining the properties of fuzzy sets and subalgebras and MP filters in logic algebra, the concepts of fuzzy subalgebra and fuzzy incident MP filter in R₀ algebra are introduced. Several equivalent characterizations for fuzzy sets to be fuzzy subalgebras are given. Some properties of fuzzy incident MP filters are discussed. It is proved that fuzzy subalgebra (fuzzy incident MP filter) is invariant under R₀ algebra homomorphism (isomorphism). Second,some basic properties of ideals and prime ideals in R₀ algebra are discussed. The lattice operations are defined on the set of all ideals in R₀ algebra M. The ideal lattice defined as above is bounded and distributive. The topological structure on the set of all prime ideals are introduced. It is proved that the set of all prime ideals is a compact T₀ topological space.In chapter five, as a counterpart of truth degree in L system, the concept of contradiction degree is introduced.Some important properties are studied.Based on these,the concept of logic equivalence is graded.And an upgrade algorithm of contradiction degree is given. Further more, difference degreeρis introduced on the set of all formulas in L system.It is proved that difference degreeρis a pseudo-metric on F(S). And some properties of pseudo-metric space (F(S),ρ) are discussed.