Shelah considered a certain version of Strong Chang's Conjecture, which we denote $\text{SCC}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted $\text{SCC}^{\text{split}}$, and prove an analogous characterization of it. In particular, $\text{SCC}^{\text{split}}$ is equivalent to the assertion that the the Friedman-Krueger poset is semiproper. This strengthens and sharpens the results of Cox, and sheds some light on problems from Usuba and Torres-Perez and Wu.

中文翻译：

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Shelah 对Strong Chang 猜想的表征的一个变体

Shelah 考虑了Strong Chang 猜想的某个版本，我们将其表示为$\text{SCC}^{\text{cof}}$，并证明了它等价于几个陈述，包括Namba forcing 是半正确的断言。我们引入了一个明显较弱的版本，表示为 $\text{SCC}^{\text{split}}$，并证明了它的类似特征。特别是，$\text{SCC}^{\text{split}}$ 等价于 Friedman-Krueger 偏序是半正确的断言。这加强并强化了 Cox 的结果，并阐明了 Usuba、Torres-Perez 和 Wu 的问题。