Motivated by certain types of ideals in pointfree functions rings, we define what we call P-sublocales in completely regular frames. They are the closed sublocales that are interior to the zero-sublocales containing them. We call an element of a frame L that induces a P-sublocale a P-element, and denote by $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) the set of all such elements. We show that if L is basically disconnected, then $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) is a frame and, in fact, a dense sublocale of L. Ordered by inclusion, the set $$\mathcal {S}_\mathfrak {p}(L)$$Sp(L) of P-sublocales of L is a complete lattice, and, for basically disconnected L, $$\mathcal {S}_\mathfrak {p}(L)$$Sp(L) is a frame if and only if $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) is the smallest dense sublocale of L. Furthermore, for basically disconnected L, $$\mathcal {S}_\mathfrak {p}(L)$$Sp(L) is a sublocale of the frame $$\mathcal {S}_\mathfrak {c}(L)$$Sc(L) consisting of joins of closed sublocales of L if and only if L is Boolean. For extremally disconnected L, iterating through the ordinals (taking intersections at limit ordinals) yields an ordinal sequence $$\begin{aligned} L\;\supseteq \;{{\,\mathrm{Pel}\,}}(L)\supseteq \;{{\,\mathrm{Pel}\,}}^2(L)\;\supseteq \;\cdots \; \supseteq \;{{\,\mathrm{Pel}\,}}^\alpha (L)\supseteq \;{{\,\mathrm{Pel}\,}}^{\alpha +1}(L)\;\supseteq \cdots \end{aligned}$$L⊇Pel(L)⊇Pel2(L)⊇⋯⊇Pelα(L)⊇Pelα+1(L)⊇⋯that stabilizes at an extremally disconnected P-frame, that we denote by $${{\,\mathrm{Pel}\,}}^\infty (L)$$Pel∞(L). It turns out that $${{\,\mathrm{Pel}\,}}^\infty (L)$$Pel∞(L) is the reflection to L from extremally disconnected P-frames when morphisms are suitably restricted.

中文翻译：

---

关于 P-Sublocales 和断开连接

受无点函数环中某些类型的理想驱动，我们在完全规则的框架中定义了我们所说的 P-sublocales。它们是包含它们的零子区域内部的封闭子区域。我们称框架 L 的一个元素为 P-元素，并用 $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) 表示所有这样的元素。我们证明如果 L 基本上是断开的，那么 $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) 是一个框架，实际上是 L 的一个密集子区域。包含，L 的 P 子区域的集合 $$\mathcal {S}_\mathfrak {p}(L)$$Sp(L) 是一个完整格，并且，对于基本上不连通的 L，$$\mathcal {S }_\mathfrak {p}(L)$$Sp(L) 是一个框架当且仅当 $${{\,\mathrm{Pel}\,}}(L)$$Pel(L) 是最小的L 的稠密子区域。此外，对于基本断开的 L，$$\mathcal {S}_\mathfrak {p}(L)$$Sp(L) 是框架 $$\mathcal {S}_\mathfrak {c}(L)$$Sc(L) 的子区域当且仅当 L 是布尔值时，由 L 的封闭子语言环境的连接组成。对于极不连接的 L，遍历序数（在极限序数处取交集）产生序数序列 $$\begin{aligned} L\;\supseteq \;{{\,\mathrm{Pel}\,}}(L) \supseteq \;{{\,\mathrm{Pel}\,}}^2(L)\;\supseteq \;\cdots \; \supseteq \;{{\,\mathrm{Pel}\,}}^\alpha (L)\supseteq \;{{\,\mathrm{Pel}\,}}^{\alpha +1}(L) \;\supseteq \cdots \end{aligned}$$L⊇Pel(L)⊇Pel2(L)⊇⋯⊇Pelα(L)⊇Pelα+1(L)⊇⋯它稳定在一个极端断开的 P 帧上，我们用 $${{\,\mathrm{Pel}\,}}^\infty (L)$$Pel∞(L) 表示。事实证明 $${{\,\mathrm{Pel}\,