## Stability of characters and filters for weighted semilattices

#### Authors

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.

DOI: 10.1007/s00233-020-10147-w

#### Status

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)

#### Reviews

[ MR4214496 (pending) | Zbl 07310708 (pending) ]

#### Abstract

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This article is based on some older unpublished work from 2015, which was temporarily integrated with a larger paper before being extracted to form a separate paper.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.

Yemon Choi