On Better-quasi-ordering Classes of Partial Orders
Gregory A. McKay
We generalise the notion of σ-scattered to partial orders and prove that some large classes of σ-scattered partial orders are better-quasi-ordered under embeddability. For every finite number 𝑛, we find an increasingly large better-quasi-ordered class of countable unions of partial orders which do not embed the rational numbers, an infinite binary tree, a dual binary tree or a reversed infinite binary tree. At 𝑛 = 2, this generalises theorems of Laver, Corominas and Thomass´e regarding σ-scattered linear orders, σ-scattered trees, countable pseudo-trees and countable 𝑁-free partial orders. In particular, a class of countable partial orders is better-quasi-ordered whenever it decomposes into a class which satisfies a natural strengthening of better-quasi-order.
Keywords: Better-quasi-order, well-quasi-order, interval, scattered, σ-scattered, partial order, structured trees