| 屈小兵,王学平.完备强对偶原子分配格上的不可约极小并分解及其应用[J].数学年刊A辑,2009,30(6):811~828 |
| 完备强对偶原子分配格上的不可约极小并分解及其应用 |
| Irredundant Minimal Join-Decompositions in Complete Distributive Lattices Which Are Strongly Coatomic and Their Applications |
| Received:September 16, 2008 Revised:December 26, 2008 |
| DOI： |
| 中文关键词: 完备格, 分配格, 强对偶原子格, 不可约极小并分解, 模糊关系方程, 极小解 |
| 英文关键词:Complete lattice, Distributive lattice, Strongly coatomic lattice, Irredundant minimal join-decomposition, Fuzzy relational equation, Minimal solution |
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| 中文摘要: |
| 在完备强对偶原子分配格上引入了不可约极小并分解的概念, 给出了元素存在不可约极小并分解的一些充要条件. 证明了当元素恰有一个下邻时, 该元素就是完全并既约元; 有两个下邻时, 元素的不可约极小并分解与不可约完全并既分解是等价的; 下邻多于两个时, 元素的不可约极小并分解不一定是不可约完全并既分解. 最后证明了模糊关系方程有极小解的充要条件是方程左边有大于等于右手项的系数或右手项系数有不可约极小并分解. |
| 英文摘要: |
| This paper introduces a concept of irredundant minimal join-decomposition in complete distributive lattices which are strongly coatomic, and gives some necessary and sufficient conditions for existence of irredundant minimal join-decomposition for an element. Then the authors show that an element with exact one lower cover is completely join irreducible, that the irredundant minimal join-decomposition of an element and its irredundant completely join-decomposition are equivalent when the number of its lower covers are equal to 2, and that the irredundant minimal
join-decomposition of an element need not be its irredundant completely join-decomposition when the number of its lower covers are more than 2. Finally, for a fuzzy relational equation it is proved that a necessary and sufficient condition for existence of its minimal solutions is whether there is a left-hand coefficient which is more than or equal to the right-hand coefficient or the right-hand coefficient has an irredundant minimal join-decomposition. |
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