格蕴涵代数的扩展LI-理想  

Extended LI-ideals in lattice implication algebras

作　　者：刘春辉[1] 

机构地区：[1]赤峰学院数学与统计学院,内蒙古赤峰024001

出　　处：《高校应用数学学报（A辑）》2015年第3期306-320,共15页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(10371106;60774073)

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?L的稳定LI-理想全体之集S(M)与L的一个LI-理想A关于任意子集M?L的扩展LI-理想全体之集EA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.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 ？ 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.

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 

分 类 号：O141.1[理学—数学] O153.1[理学—基础数学]