A little more on ideals associated with sublocales

Document Type : Research Paper


1 Department of Mathematics, Chapman University, P.O. Box 92866, California, U.S.A.

2 Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa.

3 Deparment of Mathematics and Physics, Technical University of Mombasa, P.O. Box 90420-80100, Mombasa, Kenya.



    As usual, let $\mathcal RL$ denote the ring of real-valued continuous functions on a completely regular frame $L$. Let $\beta L$ and  $\lambda L$ denote the  Stone-\v{C}ech compactification of $L$ and the Lindel\"of coreflection of $L$, respectively. There is a natural way of associating with each sublocale of $\beta L$ two ideals of $\mathcal RL$, motivated by a similar situation in $C(X)$. In~\cite{DS1}, the authors go one step further and associate with each sublocale  of $\lambda L$ an ideal of $\mathcal RL$ in a manner similar to one of the ways one does  it for sublocales of $\beta L$.  The intent in this paper is to augment~\cite{DS1} by considering two other coreflections; namely, the realcompact and the paracompact   coreflections.\\
        We show that $\boldsymbol M$-ideals of $\mathcal RL$ indexed by sublocales of $\beta L$ are precisely the intersections of maximal ideals of  $\mathcal RL$. An $\boldsymbol{M}$-ideal of $\mathcal RL$ is \emph{grounded} in case it is of the form $\boldsymbol{M}_S$ for some sublocale $S$ of $L$. A similar definition is given for an  $\boldsymbol{O}$-ideal of $\mathcal RL$.  We characterise $\boldsymbol M$-ideals of $\mathcal RL$ indexed by spatial sublocales of $\beta L$, and $\boldsymbol O$-ideals of $\mathcal RL$ indexed by closed sublocales of $\beta L$ in terms of grounded maximal ideals of $\mathcal RL$.


Main Subjects

[1] Ball, R.N. and Walters-Wayland, J., C- and C∗-quotients in pointfree topology, Dissert. Math. (Rozprawy Mat.) 412 (2002), 62 pp.
[2] Banaschewski, B., “The real numbers in pointfree topology”, Textos de Matem´atica S´erie B, No. 12, Departamento de Matem´atica da Universidade de Coimbra, 1997.
[3] Banaschewski, B. and Gilmour, C., Stone– ˇ Cech compactification and dimension theory for regular σ-frames, J. Lond. Math. Soc. 39(2) (1989), 1-8.
[4] Banaschewski, B. and Gilmour, C., Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin 37 (1996), 579-589.
[5] Banaschewski, B. and Gilmour, C., Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395-417.
[6] Banaschewski, B. and Pultr, A., Paracompactness revisited, Appl. Categ. Structures 1 (1993), 181-190.
[7] Dube, T., Notes on pointfree disconnectivity with a ring-theoretic slant, Appl. Categ. Structures 18 (2010), 55-72.
[8] Dube, T., A broader view of the almost Lindel¨of property, Algebra Universalis 65 (2011), 63–276.
[9] Dube, T., Concerning P-sublocales and disconnectivity, Appl. Categ. Structures 27 (2019), 365-383.
[10] Dube, T., On the maximal regular ideal of pointfree function rings, and more, Topology Appl. 273 (2020), 106960.
[11] Dube, T. and Stephen, D.N., On ideals of rings of continuous functions associated with sublocales, Topology Appl. 284 (2020), 107360.
[12] Dube, T. and Stephen, D.N., Mapping ideals to sublocales, Appl. Categ. Structures 29 (2021), 747-772.
[13] Gillman, L. and Jerison, M., “Rings of Continuous Functions”, Van Nostrand, Princeton, 1960.
[14] Johnson, D.G. and Mandelker, M., Functions with pseudocompact support, Gen. Topology Appl. 3 (1971), 331-338.
[15] Johnstone, P.T., “Stone Spaces”, Cambridge University Press, 1982.
[16] Madden, J. and Vermeer, J., Lindel¨of locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480.
[17] Picado, J. and Pultr, A., “Frames and Locales: topology without points”, Frontiers in Mathematics, Springer, 2012