# Stability of characters and filters for weighted semilattices

## TL;DR

This paper investigates the stability of filters in weighted semilattices, providing a combinatorial condition for the AMNM property, and demonstrates cases where this property fails for certain union-closed set systems.

## Contribution

It introduces a new combinatorial criterion for filter stability in weighted semilattices, simplifying proofs and extending the verification of the AMNM property.

## Key findings

- A necessary and sufficient condition for filter stability
- Simplified proofs of existing results
- Construction of weights where AMNM fails in certain semilattices

## Abstract

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.