超越ℵ₁的弗赖塞类的科恩力迫
Cohen Forcing with Fraïssé Classes Beyond $\aleph_1$
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中文总结 AI 辅助
本文针对超越ℵ₁的弗赖塞类，研究用有限元力迫与科恩力迫的等价性，证明无三角图类在|X|≥ℵ₂时不满足等价性，结合科恩代数特征得到具(n+1)-不并合性质的类满足等价的正面结果。
中文摘要 AI 辅助
Kostana[5]提出问题：类𝒦需满足何种自然条件，才能保证在基集X上用𝒦的有限元进行力迫与科恩力迫等价。本文证明，若𝒦_△为有限无三角图类且|X|≥ℵ₂，则𝔽_X(𝒦_△)不等价于𝔽[X]。随后利用Milovich给出的科恩代数特征得到正面结果：若ℵₙ≤|X|且𝒦具有(n+1)-不并合性质，则𝔽_X(𝒦)是科恩代数。
英文摘要
Kostana [5] asked for natural conditions on a class $\mathcal{K}$ to ensure that forcing with finite members of $\mathcal{K}$ on an underlying set $X$ is forcing equivalent to Cohen forcing. We show that if $\mathcal{K}_{\triangle}$ is the class of finite triangle-free graphs and $|X|\geq\aleph_2$, then $ \mathbb{C}_X(\mathcal{K}_{\triangle}) \not\simeq \mathbb{C}[X]. $ We then use a characterization of Cohen algebras due to Milovich to obtain a positive result. We show that if $\aleph_n\leq|X|$ and $\mathcal{K}$ has $(n+1)$-disjoint amalgamation, then $\mathbb{C}_X(\mathcal{K})$ is Cohen.
