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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2014, Vol. 29 Issue (4): 483-496    DOI:
    
The structures of compactly generated lattices in which every element has a cover
ZUO Kai, WANG Xue-ping
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
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Abstract  This paper deals with two unsolved problems raised by Dilworth and Crawley in 1973. The first one is about the existence of irredundant completely meet decomposition for an element in compactly generated lattice which is not the upper semimodular lattice. The second one is about the structure of compactly generated lattice which is not the strongly atomic lattice. This paper first proves that every element which has a cover in the compactly generated lattice has an irredundant completely meet decomposition. Finally, the concepts of locally strong modular and locally strong distributive lattice are introduced. By investigating the structures of such lattices, this paper partly comes to the conclusion to the second problem.

Key words： compactly generated lattice      semimodular lattice      atomic lattice      locally strongly modular lattice      irredundant completely meet decomposition      independent set     
Received: 12 February 2014      Published: 08 June 2018
CLC:  O153.1  
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ZUO Kai
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ZUO Kai, WANG Xue-ping. The structures of compactly generated lattices in which every element has a cover. Applied Mathematics A Journal of Chinese Universities, 2014, 29(4): 483-496.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2014/V29/I4/483


每个元有上覆盖的紧生成格的结构

Dilworth与Crawley1973年提出能否去掉上半模格条件来刻画元素的不可约完全交既分解问题以及能否去掉强原子格的条件刻画紧生成格结构的问题, 本文首先证明了每个元有上覆盖的紧生成格$L$中任意元有不可约完全交既分解, 从而肯定地回答了Dilworth与Crawley上述第一个问题. 之后, 在每个元有上覆盖的紧生成格中引入局部强模格与局部强分配格的概念, 研究了局部强模格中独立集的特性以及局部强模格与局部分配格的结构, 从而部分解决了Dilworth与Crawley上述第二个问题.

关键词： 紧生成格,  局部强模格,  上半模格,  原子格,  不可约完全交既分解,  独立集 
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