非可换逻辑代数的滤子及模糊化理论

【作者】 王伟；

【导师】 辛小龙；

【作者基本信息】 西北大学 ， 基础数学， 2010， 博士

【摘要】 非可换逻辑代数是非可换逻辑相应的代数结构.BL代数是基本逻辑(Basic Logic)系统的代数结构,而伪BL代数是BL代数的非可换推广.非可换剩余格又是伪BL代数、伪MTL代数等的推广.我们在研究工作中,发现上述非可换逻辑代数结构可以通过滤子及模糊滤子来进行研究和刻画.因而,本文主要研究了几类非可换逻辑代数中的滤子及模糊滤子的理论,为非可换逻辑代数的深入研究奠定了坚实基础.我们所做的具体工作如下：1.引入并讨论了伪BL代数的模糊滤子、模糊素滤子、模糊布尔滤子、模糊正规滤子、模糊超滤子和模糊固执滤子,研究了相应模糊滤子的性质.由分配格中Stone模糊素理想定理启发,我们在伪BL代数中建立模糊素滤子定理.通过讨论几类模糊滤子之间的关系,特别是模糊布尔滤子和模糊正规滤子的关系,我们解决了一个公开问题,即“在伪BL代数中是否每一个布尔滤子都是正规的.”2.引入并讨论了伪BL代数的超滤子和固执滤子,讨论了伪BL代数的布尔滤子和正规滤子的关系,采用非模糊化的途径给出了上述公开问题的一种新的解决方法.3.我们将上述有关工作推广到了剩余格,引入并讨论了剩余格的模糊滤子、模糊素滤子、模糊布尔滤子、模糊正规滤子、模糊超滤子和模糊固执滤子,研究了相应模糊滤子的性质.我们在剩余格中建立模糊素滤子定理.通过讨论几类模糊滤子之间的关系,特别是模糊布尔滤子和模糊正规滤子的关系,我们在剩余格中研究了相应的问题.4.引入了剩余格的伪G滤子、伪MV滤子、超滤子和固执滤子,研究了相应的性质,特别是研究了布尔滤子和正规滤子的关系.5.引入了Heyting代数的布尔滤子、素滤子、蕴含滤子、正蕴含滤子、超滤子和固执滤子,定义了Heyting代数的模糊滤子、模糊布尔滤子、模糊素滤子、模糊蕴含滤子、模糊正蕴含滤子、模糊超滤子和模糊固执滤子,并讨论了其相应的性质.进一步,我们证明在Heyting代数中,模糊布尔滤子等价于模糊蕴含滤子,模糊正蕴含滤子等价于模糊滤子,并讨论了这几类滤子之间的关系.最后,我们研究了Heyting代数的直觉模糊滤子,讨论了相应性质,证明了Heyting代数上的上的直觉模糊格滤子等价于直觉模糊滤子.更多还原

【Abstract】 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.更多还原