Unavoidable subprojections in union-closed set systems of infinite breadth

摘要

We obtain two combinatorial structure theorems for semilattices of infinitebreadth, exploiting their representation as union-closed set systems. The firstof these shows that every such semilattice admits a subquotient isomorphic toone of three natural examples. The second has a Ramsey-theoretic flavour: westudy how a union-closed set system interacts with a given decomposition of theunderlying set, and show that there is either extreme fragmentation or somecontrolled structure. The structure theorems are used to study a stability problem for filters insemilattices, relative to a given weight function: we show that when aunion-closed set system has infinite breadth, one may construct a weightfunction on it such that stability of filters fails. This has applications tothe study of character stability for Banach algebras.