Concerning P-Sublocales and Disconnectivity

Abstract

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 ${{\,\mathrm{Pel}\,}}(L)$ the set of all such elements. We show that if L is basically disconnected, then ${{\,\mathrm{Pel}\,}}(L)$ is a frame and, in fact, a dense sublocale of L. Ordered by inclusion, the set $\mathcal {S}_\mathfrak {p}(L)$ of P-sublocales of L is a complete lattice, and, for basically disconnected L, $\mathcal {S}_\mathfrak {p}(L)$ is a frame if and only if ${{\,\mathrm{Pel}\,}}(L)$ is the smallest dense sublocale of L. Furthermore, for basically disconnected L, $\mathcal {S}_\mathfrak {p}(L)$ is a sublocale of the frame $\mathcal {S}_\mathfrak {c}(L)$ 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

$$\begin{aligned} L\;\supseteq \;{{\,\mathrm{Pel}\,}}(L)\supseteq \;{{\,\mathrm{Pel}\,}}^2(L)\;\supseteq \;\cdots \; \supseteq \;{{\,\mathrm{Pel}\,}}^\alpha (L)\supseteq \;{{\,\mathrm{Pel}\,}}^{\alpha +1}(L)\;\supseteq \cdots \end{aligned}$$

that stabilizes at an extremally disconnected P-frame, that we denote by ${{\,\mathrm{Pel}\,}}^\infty (L)$. It turns out that ${{\,\mathrm{Pel}\,}}^\infty (L)$ is the reflection to L from extremally disconnected P-frames when morphisms are suitably restricted.