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