A classical theorem of Hechler asserts that the structure $$\left( \omega ^\omega ,\le ^*\right) $$ ω ω , ≤ ∗ is universal in the sense that for any $$\sigma $$ σ -directed poset $${\mathbb {P}}$$ P with no maximal element, there is a ccc forcing extension in which $$\left( \omega ^\omega ,\le ^*\right) $$ ω ω , ≤ ∗ contains a cofinal order-isomorphic copy of $${\mathbb {P}}$$ P . In this paper, we prove the following consistency result concerning the universality of the higher analogue $$\left( \kappa ^\kappa ,\le ^S\right) $$ κ κ , ≤ S : assuming $$\textsf {GCH }$$ GCH , for every regular uncountable cardinal $$\kappa $$ κ , there is a cofinality-preserving $$\textsf {GCH }$$ GCH -preserving forcing extension in which for every analytic quasi-order $${\mathbb {Q}}$$ Q over $$\kappa ^\kappa $$ κ κ and every stationary subset S of $$\kappa $$ κ , there is a Lipschitz map reducing $${\mathbb {Q}}$$ Q to $$(\kappa ^\kappa ,\le ^S)$$ ( κ κ , ≤ S ) .

中文翻译：

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包含模非平稳

Hechler 的一个经典定理断言 $$\left(\omega ^\omega ,\le ^*\right) $$ ω ω , ≤ ∗ 是通用的，因为对于任何 $$\sigma $$ σ -有向poset $${\mathbb {P}}$$ P 没有最大元素，存在一个ccc 强制扩展，其中$$\left( \omega ^\omega ,\le ^*\right) $$ ω ω , ≤ ∗ 包含 $${\mathbb {P}}$$ P 的同构同构副本。在本文中，我们证明了以下关于更高类似物 $$\left( \kappa ^\kappa ,\le ^S\right) $$ κ κ , ≤ S 的普遍性的一致性结果：假设 $$\textsf {GCH }$$ GCH ，对于每一个常规的不可数基数 $$\kappa $$ κ ，都有一个保留共终结性的 $$\textsf {GCH }$$ GCH 保留强制扩展，其中对于每个解析准序 $${\ mathbb {Q}}$$ Q 超过 $$\kappa ^\kappa $$ κ κ 和 $$\kappa $$ κ 的每个平稳子集 S，