In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset has a nested chain decomposition. This conjecture is proved to be true only for some posets of small ranks (Wang Discrete Math. 145(3), 493–497, 2005; Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011). In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfy the Griggs’s conjecture by refining the proofs in the two papers (Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011).

中文翻译：

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秩3归一化匹配组中嵌套的若干注解

1970 年代，Griggs 推测每个归一化匹配秩单峰偏序集都有一个嵌套链分解。这个猜想仅对一些小秩的偏序集成立（Wang Discrete Math. 145(3), 493–497, 2005; Hsu et al. Discrete Math. 309(3), 521-531, 2009; Escamilla et al. Order 28, 357–373, 2011)。在本文中，我们通过改进两篇论文中的证明（Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla 等人，第 28 号命令，357–373，2011 年）。