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}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain.

中文翻译：

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SCOTT 域上的 WADGE 订单不是一个很好的准订单

我们证明了 Scott 域的 Borel 子集上的 Wadge 阶不是一个准准阶，并且这个特征甚至出现在最多为 2 的 Borel 秩的集合中。为此，一个特定类的可数 2-彩色的$\mathbb{P}_{emb} $由同态诱导的阶嵌入到 Wadge 阶中$\三角洲_2^0 $-Scott 域的度数。然后我们证明$\mathbb{P}_{emb} $就这种比较概念而言，允许无限严格递减链和无限反链，因此转移到$\三角洲_2^0 $-Scott 域的度数。