WBR₀-代数的等价刻画及其弱化

【作者】 王娜；

【导师】 吴洪博；

【作者基本信息】 陕西师范大学 ， 基础数学， 2013， 硕士

【摘要】 多值逻辑的一个重要研究方向是对有关代数系统的研究,不同的多值逻辑系统对应着不同的多值逻辑代数.著名逻辑学家C.C.Chang提出了MV-代数,此后,吴望明教授提出了FI-代数,徐扬教授建立了格蕴涵代数,P.Hajek教授建立了BL-代数,王国俊教授建立了独立于BL-代数的R0-代数.吴洪博教授通过对R0-代数的研究提出了BR0-代数以及WBR0-代数.本文对WBR0-代数进行了进一步的研究.本文的结构和详细内容安排如下：第1章预备知识.本章给出了文章中将要用到的FI-代数,WBR0-代数,偏序集,Pt-模(偏序集上的t-模)和Ps-模(偏序集上的s-模)的基本概念.第2章WBR0-代数的Pt-模和Ps-模表示.首先,给出了WBR0-代数的Pt-模(偏序集上的t-模)的表现形式；其次,在偏序集上通过Pt-模结合蕴涵算子给出了WBR0-代数的表示形式；最后,证明了WBR0-代数原始定义中的二元运算(?)是WBR0-代数的Ps-模(偏序集上的s-模),并通过Ps-模结合蕴涵算子于偏序集上给出了WBR0-代数的表示形式.第3章WBR0-代数的两种弱化形式及性质.首先将WBR0-代数的条件进行了两种不同形式的弱化,建立了模糊BR0-代数(FBR0-代数)结构和正则BR0-代数(RBR0-代数)结构,并讨论了其中的相关性质.然后,证明了正则BR0-代数是模糊BR0-代数；模糊BR0-代数和FI-代数是相同代数结构；正则BR0-代数和正则FI-代数是相同代数结构.最后,分别构造了非正则BR0-代数的模糊BR0-代数和非WBR0-代数的正则BR0-代数,从而阐明了模糊BR0-代数,正则BR0-代数,WBR0-代数的相互蕴涵关系和相对独立性.更多还原

【Abstract】 Studying algebraic system is an important research direction of many-valued logic, different many-valued logic systems correspond to different many-valued log-ic algebras. After famous logician C.C.Chang proposed MV-algebras, Professor Wu Wangming defined FI-algebras, Professor Xu Yang proposed Lattice implication algebras, Professor P.Hajek established BL-algebras, Professor Wang Guojun estab-lished R0-algebras which are different from BL-algebras. Through the research about R0-algebras by Professor Wu Hongbo, BR0-algebras and WBR0-algebras have been introduced. In this paper, the WBR0-algebras have been studied in depth.the construction and the detailed contents of this paper are as follows:Chapter1:Preliminaries. In this chapter,we give the basic concepts of FI-algebras, WBR0-algebras, posets, Pt-norms (t-norms on posets) and Ps-norms (s-norms on posets) which will be used in this paper.Chapter2:Representation of WBR0-algebras by Pt-norms and Ps-norms. Firstly, the form of Pt-norms (t-norms on posets) on WBRo-algebras is given; Sec-ondly, the equivalence representation of WBR0-algebras has been obtained on posets though Pt-norms and implication operators; Finally, it is proved that the binary op-eration (?) in the original definition of the WBR0-algebras is the Ps-norms(s-norm on posets) of WBR0-algebras, and the equivalence representation of WBR0-algebras has been obtained on posets though Ps-norms and implication operators.Chapter3:The2-kinds of weak forms of WBR0-algebras and their proper-ties. First of all, by weakening the conditions of the WBRo-algebras with2-kinds of different forms, the fuzzy BR0-algebras (FBR0-algebras) and the regular BRo-algebras (RBR0-algebras) are built and then their properties are discussed. Then, it is proved that RBR0-algebras are FBR0-algebras, FBR0-algebras have the same algebraic structure with FI-algebras and RBR0-algebras have the same algebraic structure with regular Fl-algebras. Finaly, it is also showed that FBR0-algebras are not RBRo-algebras and RBRo-algebras are not WBR0-algebras by constructing a example respectively, which explains that there exist the implicational relation and relative independence between3-kinds of algebras.更多还原