Ordered sets with no chains of ideals of a given type

摘要

We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type α, there is a finite listA1α, ...,Anαof ordered sets such that for every ordered setPthe setJ(P)of ideals ofP, ordered by inclusion, contains a chain of type α if and only ifPcontains a subset isomorphic to one of theA1#x03B1;, ...,Anα. The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by L