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高校应用数学学报
高校应用数学学报  2015, Vol. 30 Issue (3): 306-320    
    
格蕴涵代数的扩展LI-理想
刘春辉
赤峰学院 数学与统计学院, 内蒙古赤峰 024001
Extended LI-ideals in lattice implication algebras
LIU Chun-hui
Department of Mathematics and Statistics, Chifeng University, Chifeng 024001, China
 全文: PDF 
摘要： LI-理想是研究格蕴涵代数结构特征的一个重要的工具性概念. 综合运用代数学与逻辑学的方法和原理对格蕴涵代数的LI-理想理论作进一步深入研究. 首先, 引入格蕴涵代数$L$的LI-理想$A$关于$L$的子集$M$的扩展LI-理想及稳定LI-理想概念并考察它们的基本性质. 其次, 讨论了$L$的几类扩展LI-理想集的格论特征.证明了$L$的关于一个给定子集$M\subseteq{L}$的稳定LI-理想全体之集$\mathscr{S}(M)$与 $L$的一个LI-理想$A$关于任意子集$M\subseteq{L}$的扩展LI-理想全体之集$\mathscr{E}_{A}$均构成完备Heyting代数的结论. 再次, 给出了商格蕴涵代数和乘积格蕴涵代数的扩展LI-理想的若干性质. 最后, 借助于$L$的扩展LI-理想之特性获得了$L$的ILI-理想的若干等价刻画.
关键词： 格值逻辑;  格蕴涵代数;  LI-理想;  扩展LI-理想;  ILI-理想;  完备Heyting代数    
Abstract: LI-ideals is an important tool for studying the structure characteristics of lattice implication algebras. In this paper, the theory of LI-ideals in lattice implication algebras was further studied by using the methods and principles of algebra and logic. Firstly, the notions of extended LI-ideals and stable LI-ideals of a LI-ideal $A$ associated to a subset $M$ of lattice implication algebra $L$ are introduced and some of their basic properties are investigated. Secondly, some lattice theory characteristics about some types sets of extended LI-ideals in a lattice implication algebra $L$ are discussed. It's proved that the set of all stable LI-ideals associated to a given subset $M\subseteq{L}$ and the set of all extended LI-ideals of LI-ideals $A$ associated to any subset of $L$ both form complete Heyting algebras. Thirdly, some properties of extended LI-ideals in quotient and product lattice implication algebras are given. Finally, some equivalent characterizations of ILI-ideals are obtained by means of extended LI-ideals in lattice implication algebras.
Key words: lattice valued logic    lattice implication algebra    LI-ideal    extended LI-ideal    ILI-ideal    complete Heyting algebra
收稿日期: 2014-09-24 出版日期: 2018-05-27
CLC:  O141.1  
基金资助: 国家自然科学基金(10371106; 60774073)
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引用本文:

刘春辉. 格蕴涵代数的扩展LI-理想[J]. 高校应用数学学报, 2015, 30(3): 306-320.

链接本文:

http://www.zjujournals.com/amjcua/CN/Y2015/V30/I3/306

[1] 刘春辉. 否定非对合剩余格的LI-理想理论[J]. 高校应用数学学报, 2015, 30(4): 445-456.