On better-quasi-ordering classes of partial orders

TL;DR

This paper introduces a new method for constructing better-quasi-orders and extends the concept of $\sigma$-scatteredness to partial orders, proving that this class is well-structured under embeddability.

Contribution

It generalizes existing theorems on $\sigma$-scattered linear orders and trees to broader classes of partial orders using a novel construction technique.

Findings

The class of $\sigma$-scattered partial orders is better-quasi-ordered under embeddability.

The method generalizes previous results on linear orders, trees, and forests.

Countable partial orders with certain indecomposable subsets are better-quasi-ordered.

Abstract

We provide a method of constructing better-quasi-orders by generalising a technique for constructing operator algebras that was developed by Pouzet. We then generalise the notion of $\sigma$-scattered to partial orders, and use our method to prove that the class of $\sigma$-scattered partial orders is better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomass\'{e} regarding $\sigma$-scattered linear orders and trees, countable forests and N-free partial orders respectively. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order.