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Concerning P-Sublocales and Disconnectivity

  • Themba DubeEmail author
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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.

Keywords

Completely regular frame \(F^\prime \)-frame Basically disconnected frame Extremally disconnected frame Sublocale P-sublocale Functor Reflective subcategory 

Mathematics Subject Classification

Primary: 06D22 Secondary: 13A15 18A22 54C30 54G05 54G10 54G15 

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Notes

Acknowledgements

Thanks are due to the referee for suggestions that have improved the paper, especially with regard to presentation.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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