【作者】 张宝华；

【导师】 徐扬；

【作者基本信息】 西南交通大学 ， 应用数学， 2003， 硕士

【摘要】 非经典逻辑是人工智能领域中十分活跃的研究方向，是不确定性推理的理论基础。在非经典逻辑的研究中，格值逻辑的研究具有重要而广泛的意义。格蕴涵代数是徐扬教授为研究格值逻辑把格与蕴涵代数相结合提出的一个代数系统。本文将在格蕴涵代数已有性质的基础上，进一步讨论格蕴涵代数的性质及结构。具体作了以下三方面的工作： 1、在有界分配格的素理想空间上给出了格蕴涵代数的一个表示，并借助集合的无限运算系统地讨论了表示的性质，得出了格蕴涵代数中任意子集无限运算的若干性质。 2、对格蕴涵代数中由一子集生成的格蕴涵子代数的结构及格蕴涵代数的直积进行了研究，给出了一格蕴涵代数可分解成一族格蕴涵代数直积的必要条件，证明了格H蕴涵代数是一有单位元的布尔环。 3、给出了格蕴涵代数中O-理想的定义并讨论了其性质，给出了格蕴涵代数中集合S上超滤子的定义并与集合代数上的超滤子作了比较，研究了两种超滤子之间的关系。更多还原

【Abstract】 Non-classical logic is an active research direction in the field of artificial intelligence and a logic foundation for uncertainty reasoning. In the study of non-classical logic, lattice-valued logic system is of extensive significance. Lattice implication algebra is an algebraic system combinating lattice with implication algebra. It is first defined by professor Xu Yang in order to study lattice-valued logic. There are many research papers about lattice implication algebra and related logic. In this paper, the proper and structure of lattice implication algebra is further studied. The following works have been done:1. On the prime ideal space of distributive lattice with bounds, a representation of lattice implication algebra was given. And with the help of the limit of set, the properties of this representation were discussed. Finally, some of properties of any subset’s limit of lattice implication algebra have been proved.2.The structure of the lattice implication sub-algebra which generated from a subset, and the product of lattice implication algebra were studied. The necessary condition was given of a lattice implication algebra which can be decomposed into lattice implication algebra’s product. It is proved that lattice H implication algebra is a Boole ring with a unit.3.The concepts of O-ideal was proposed and its property was discussed. Ultra-filter on a set S of lattice implication algebra was defined and was compared with the ultra-filter on a set of lattice, the relationship between this two ultra-filter was studied.更多还原