Non-filter-for-logic Algebra And Fuzzy Theory

The non-commutative logical algebras are the algebraic counterpart of the non-commutative logic. BL-algebras are algebraic structure of Basic Logic System. The pseudo BL-algebras are non-commutative extension of BL-algebras and the non-commutative residuated lattices are extension of pseudo BL-algebras and pseudo MTL-algebras. In our research work, we find the structure of the above non-commutative logical algebras can be de-scribed and researched by the tools of filters and fuzzy filters. Therefore, in this dissertation, the theory of filters and fuzzy filters in several kinds of non-commutative logical algebras are studied, which lays a good foundation for the forthcoming deep research in non-commutative logical algebras. Details are as follows:1. The concepts of fuzzy filters, fuzzy prime filter, fuzzy Boolean fil-ter, fuzzy normal filter, fuzzy ultrafilter and fuzzy obstinate filter of pseudo BL-algebras are introduced and discussed, and the properties of them are studied. Inspired by the Stone Fuzzy Prime Ideal Theory in distributive lat-tice, we construct the Fuzzy Prime Fuzzy Theory in pseudo BL-algebras. By discussing the relations among the above fuzzy filters, especially the relation between fuzzy Boolean filter and fuzzy normal filter, we solve an open prob-lem that "Prove or negate that every Boolean filter is normal in pseudo-BL algebras."2. The concepts of ultrafilter and obstinate filter of pseudo BL-algebras are introduced and discussed and the relation between Boolean filter and normal filter of pseudo BL-algebras is discussed, by which a new solution for the open problem is found with the method of non-fuzzification.3. The above research work can be applied to the residuated lattices. The concepts of fuzzy filters, fuzzy prime filter, fuzzy Boolean filter, fuzzy normal filter, fuzzy ultrafilter and fuzzy obstinate filter of residuated lattices are introduced and discussed, and the corresponding properties of them are studied. The Fuzzy Prime Fuzzy Theory in residuated lattices is constructed. By discussing the relations among the above fuzzy filters, especially the re-lation between fuzzy Boolean filter and fuzzy normal filter, we studied the counterpart problem in residuated lattices.4. The concepts of pseudo G-filters, pseudo MV-filter, ultrafilter and obstinate filter of residuated lattices are introduced and discussed, and the corresponding properties of them, especially the relation between Boolean filter and normal filter are studied.5. The concepts of Boolean filter, prime filter, implicative filter, posi-tive implicative filter, ultrafilter and obstinate filter of Heyting algebras are introduced. Then fuzzy filters, fuzzy Boolean filter, fuzzy prime filter, fuzzy implicative filter, fuzzy positive implicative filer, fuzzy ultrafilter and fuzzy obstinate filter of Heyting algebras are defined and the corresponding prop-erties of them are discussed. Furthermore, it is proved that in Heyting al-gebras, fuzzy Boolean filter is equivalent to fuzzy implicative filter, fuzzy positive implicative filer is equivalent to fuzzy filter, and the relations among the above fuzzy filters are discussed. Finally, the intuitionistic fuzzy filter of Heyting algebras is studied, and the corresponding properties of it is stud-ied. Then it is proved that in Heyting algebras, the intuitionistic fuzzy filter is equivalent to the intuitionistic fuzzy filter.