The Wadge order on the Scott domain is not a well-quasi-order

TL;DR

This paper demonstrates that the Wadge order on Borel subsets of the Scott domain is not a well-quasi-order by embedding a complex poset structure, revealing infinite decreasing chains and antichains even at low Borel ranks.

Contribution

It introduces a specific class of countable 2-colored posets and embeds them into the Wadge order, proving the existence of infinite chains and antichains in the Wadge degrees of the Scott domain.

Findings

Wadge order on Borel subsets of the Scott domain is not a well-quasi-order

Existence of infinite strictly decreasing chains in the Wadge order

Existence of infinite antichains in the Wadge order

Abstract

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{lay}$ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\boldsymbol{\Delta}^0_2$-degrees of the Scott domain. We then show that $\mathbb{P}_{lay}$ both admits infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\boldsymbol{\Delta}^0_2$-degrees of the Scott domain.