Relating Subsets of a Poset, and a Partition Theorem for WQOs

This note collects together some observations I made in the context of comparing graph properties by the graph minor relation [ 2 ], but which apply more generally to arbitrary subsets of a given poset. They are all simple and ought to be well known, but I have been unable to find a source. The main observation is that the subsets of an infinite poset P can be decomposed in a way that resembles factoring: there are ‘indivisible’ sets that behave like primes, and if P is a WQO then it factors uniquely into such indivisible sets. Let (P, ) be any poset, typically infinite. (Countable will do to make things intersting, and a particularly interesting case will be that P is a WQO.) Given subsets A,B ⊆ P , let us write A B to express that for every a ∈ A there is a b ∈ B such that a b. This is a quasi-ordering on the power set of P , which induces a partial ordering on the set P of ∼-equivalence classes of subsets of P , where A ∼ B if A B and B A. Although we shall continue to speak about the subsets themselves rather their equivalence classes, we shall often distinguish them only up to equivalence to derive properties of this poset P. For A,B ⊆ P we write A < B if A B but A ∼ B (ie. B A). The first problem we address is whether we can always find particularly typical representatives of these equivalence classes, in the following sense. Given an infinite set A ⊆ P , we can obtain numerous equivalent sets just by ‘adding junk’: for every A′ < A we clearly have A∪A′ ∼ A. This process is not easily reversible: if we are given A∪A′ as a single set, we may not be able to identify and discard its ‘inessential’ part A′. So it seems that ‘lean’ sets not containing large amounts of such junk are particularly desirable representatives of their equivalence classes. To make this precise, let us call an infinite set A ⊆ P lean if A A′ for every A′ ⊆ A with |A′| = |A|. (For example, the set of finite stars and the set of finite paths are both lean under the minor relation for graphs, but the set of finite trees is not lean.) Then our first question is: which infinite sets A ⊆ P are equivalent to some lean set A′ ⊆ P , possibly with A′ ⊆ A? Clearly some are not: the union U of the set of finite stars and the set of finite paths, for instance, is not equivalent to any lean set of finite graphs. The