A note on well quasi-orderings for powersets

Abstract

This note characterizes those quasi-orderings (A,⪯) for which ( P(A),⊑) are well quasi-orderings, where B 1⊑B 2 iff (∀y∈B 2)(∃x∈B 1): x⪯y (for B 1,B 2⫅A ). It turns out that they are those which do not contain the “Rado structure”, hence are ω 2 -well quasi-orderings in other words. A motivation for the question has come from the area of verification of infinite-state systems, where the usefulness of well quasi-orderings has already been recognized. This note suggests that finer notions might be useful as well. In particular, ω 2 -well quasi-orderings illuminate a specific problem related to termination of a reachability algorithm, which has been touched on by Abdulla and Jonsson.