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高校应用数学学报
高校应用数学学报  2014, Vol. 29 Issue (4): 475-482    
    
格蕴涵代数的LI-理想格及其素元刻画
刘春辉
赤峰学院 数学与统计学院, 内蒙古赤峰 024001
LI-ideals lattice and its prime elements characterizations in a lattice implication algebra
LIU Chun-hui 
Department of Mathematics and Statistics, Chifeng University, Chifeng 024001, China
 全文: PDF 
摘要： 运用格理论的原理和方法对格蕴涵代数$L$的LI-理想概念作进一步研究. 首先, 在$L$的全体LI-理想之集$\mathscr{I}_{LI}(L)$上定义了格运算$\Cap$和$\Cup$, 蕴涵运算$\Longrightarrow$以及伪补运算$\circledast$, 证明了$(\mathscr{I}_{LI}(L), \subset, \Cap, \Cup, \Longrightarrow, \{O\}, L)$构成一个完备Heyting代数的结论. 其次, 利用运算$\circledast$的性质给出了$(\mathscr{I}_{LI}(L), \subset, \Cap, \Cup, \Longrightarrow, \circledast, \{O\}, L)$成为Boolean代数的若干充要条件. 最后, 借助于$L$的素LI-理想之特性获得了格$(\mathscr{I}_{LI}(L), \subset, \Cap, \Cup, \{O\}, L)$中素元的若干等价刻画.
关键词： 格值逻辑;  格蕴涵代数;  LI-理想;  完备Heyting代数;  Boolean代数;  素元    
Abstract: The concept of LI-ideals in lattice implication algebras $L$ was further studied by using the principle and method of lattice theory. Firstly, the lattice operations $\Cap$ and $\Cup$, implication operation $\Longrightarrow$ and pseudo-complement operation $\circledast$ are defined on the set $\mathscr{I}_{LI}(L)$ which contains all LI-ideals in $L$, and $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \Longrightarrow, \{O\}, L)$ is proved to form a complete Heyting algebra. Secondly, some necessary and sufficient conditions of $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \Longrightarrow, \circledast, \{O\}, L)$ becoming a Boolean algebra are given by using properties of operation $\circledast$. Finally, some equivalent characterizations of prime elements in lattice $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \{O\}, L)$ are obtained by means of prime LI-ideals in $L$.
Key words: lattice valued logic    lattice implication algebra    LI-ideal    complete Heyting algebra    Boolean algebra    prime element
收稿日期: 2014-04-28 出版日期: 2018-06-08
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LI-理想格及其素元刻画[J]. 高校应用数学学报, 2014, 29(4): 475-482.

LIU Chun-hui. LI-ideals lattice and its prime elements characterizations in a lattice implication algebra. Applied Mathematics A Journal of Chinese Universities, 2014, 29(4): 475-482.

链接本文:

http://www.zjujournals.com/amjcua/CN/    或    http://www.zjujournals.com/amjcua/CN/Y2014/V29/I4/475

[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.
[2] 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.