We continue the study of the AMNM property for weighted semilattices that was initiated in [Y. Choi, J. Austral. Math. Soc. 95 (2013), no. 1, 36-67; arXiv 1203.6691 ] . We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this "filter stability" property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in [Choi, ibid. ] , and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.

中文翻译：

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加权半格的字符和滤波器的稳定性

我们继续研究在 [Y. Choi, J. Austral。数学。社会。95 (2013)，没有。1、36-67；arXiv 1203.6691]。我们根据过滤器相对于给定权重函数的稳定性重新表述了这一点，然后提供了一个组合条件，该条件对于保持这种“过滤器稳定性”属性是必要和充分的。给出的例子表明，这个新条件允许更容易和统一地证明 [Choi，同上。] ，此外还允许我们在该论文结果未涵盖的情况下验证 AMNM 属性。作为最后的应用，我们展示了对于一大类作为联合闭集系统自然产生的半格，人们总是可以构造 AMNM 属性失败的权重。