Yemon Choi*, Mahya Ghandehari, Hung Le Pham
* Corresponding author
Keywords: AMNM, breadth, Hyers-Ulam stability, semilattice, stable characters, stable filters
MSC 2020: Primary 06A12. Secondary 43A22.
Appeared as Semigroup Forum 102 (2021), no. 1, 86–103 (open access)
Preprint version available at arXiv 1901.00082v3 (see remarks below regarding version history)
[ MR4214496 (pending) | Zbl 07310708 (pending) ]
We continue the study of the AMNM property for weighted semilattices that was initiated in [Y. Choi, J. Austral. Math. Soc. 2013]. 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.
This article is based on some older unpublished work from 2015, which was temporarily integrated with a larger paper before being extracted to form a separate paper.
Although the article was written in 2020, it was posted to arXiv not as a new submission but as a replacement to an earlier arXiv posting, in line with arXiv recommendations. Version 1 of that posting should therefore be disregarded.