Inclusion modulo nonstationary

Joint work with Gabriel Fernandes and Miguel Moreno.

Abstract. A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset $\mathbb P$ with no maximal element, there is a ccc forcing extension in which $\left(\omega^\omega,\le^*\right)$ contains a cofinal order-isomorphic copy of $\mathbb P$.

In this paper, we prove a consistency result concerning the universality of the higher analogue $(\kappa^\kappa,\le^S)$.

Theorem. Assume GCH. For every regular uncountable cardinal $\kappa$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order $\mathbb Q$ over $\kappa^\kappa$ and every stationary subset $S$ of $\kappa$, there is a Lipschitz map reducing $\mathbb Q$ to $(\kappa^\kappa,\le^S)$.

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Citation information:

G. Fernandes, M. Moreno and A. Rinot, Inclusion modulo nonstationary, Monatsh. Math., 192(4): 827-851, 2020.