单边FS假设下的斯科特函数空间：反例、肯定结果与新方向

中文摘要 AI 辅助

已知当源和目标都是FS域时，FS域类在斯科特函数空间下是封闭的。本文研究单边FS假设下的情况，特别强调普洛特金纽带的作用。我们建立了两个互补的连续性定理。首先，当\(X\)是FS域时，斯科特函数空间\([X→T]\)是连续dcpo，证明引入了普洛特金纽带的有限层截断映射。其次，当\(L\)是FS域时，斯科特函数空间\([T→L]\)也是连续dcpo，论证基于有限分离近似恒等式及普洛特金纽带双分支序结构的有限控制分析。为确定这些肯定结果的极限，考虑劳森闭圆盘域，虽\(\Disk^{\top}\)是FS域，但\([\Disk^{\top}→T]\)是连续的却非FS域。本文最后确定了现有方法的边界并提出统一近似原理，可能为斯科特函数空间的连续性提供一般刻画。

英文摘要

The class of FS-domains is known to be closed under Scott function spaces when both the source and target are FS-domains. This paper investigates what remains true under one-sided FS assumptions, with particular emphasis on the role of Plotkin's tie. We establish two complementary continuity theorems. First, whenever \(X\) is an FS-domain, the Scott function space \([X\to T]\) is a continuous dcpo. The proof introduces finite-layer truncation maps on Plotkin's tie, which generate directed families of way-below approximants below every Scott-continuous map. Secondly, whenever \(L\) is an FS-domain, the Scott function space \([T\to L]\) is again a continuous dcpo. Here the argument is based on finitely separating approximate identities, together with a finite-control analysis of the two-branch order structure of Plotkin's tie. These two approximation mechanisms are conceptually different but both produce the directed families of way-below approximants required for continuity. To determine the limits of these positive results, we consider the Lawson closed-disk domain. Although \(\Disk^{\top}\) is an FS-domain, the Scott function space \([\Disk^{\top}\to T]\) is shown to be continuous but not itself an FS-domain. This establishes that preservation of continuity is strictly weaker than preservation of the FS property. The paper concludes by identifying the boundaries of the present methods and proposing a unified approximation principle that may provide a general characterization of continuity for Scott function spaces.

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31 pages, 2 figures