稳定连续偏序集的紧-开对偶性
Compact-Open Dualities for Stably Continuous Posets
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中文总结 AI 辅助
本研究梳理推广连续偏序集的对偶关系，证明含稳定性、完备性等指标的核心对偶定理，定义梯级指标概念，并探讨代数性、邻近格与完美映射。
中文摘要 AI 辅助
我们梳理并推广了若干涉及连续偏序集的对偶关系。核心定理为$\n\mathbf{St}_α\mathbf{Inf}_{α'}\mathbf{Cont}_{β'}\mathbf{Sup}_β\simeq (\mathbf{St}_β\mathbf{Inf}_{β'}\mathbf{Cont}_{α'}\mathbf{Sup}_α)^{\mathrm{op}}$，其中$\mathbf{St}$、$\mathbf{Inf}$、$\mathbf{Cont}$和$\mathbf{Sup}$分别指稳定性、完备性、连续性和余完备性。指标为“梯级”，即对依赖和与商封闭的集合类$λ$，配有相关的$λ$-小下确界和$λ$-滤过上确界概念。论文后半部分讨论了代数性、邻近格和完美映射。
英文摘要
We organize and generalize several dualities involving continuous posets. The main theorem reads $\mathbf{St}_α\mathbf{Inf}_{α'}\mathbf{Cont}_{β'}\mathbf{Sup}_β\simeq (\mathbf{St}_β\mathbf{Inf}_{β'}\mathbf{Cont}_{α'}\mathbf{Sup}_α)^{\mathrm{op}}$, where $\mathbf{St}$, $\mathbf{Inf}$, $\mathbf{Cont}$ and $\mathbf{Sup}$ refer to stability, completeness, continuity and cocompleteness. The indices are "ladders", i.e., classes of sets $λ$ stable under dependent sums and quotients, with associated notions of $λ$-small infima and $λ$-filtered suprema. In the second half of the paper, we discuss algebraicity, proximity lattices and perfect maps.
