A little more on ideals associated with sublocales

Document Type : Research Paper

Authors

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.

10.48308/cgasa.2023.234093.1456

Abstract

    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 $\mathcal RL$.

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References

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Categories and General Algebraic Structures with Applications
Volume 20, Issue 1
Special Issue Dedicated to Prof. Themba Dube
January 2024
Pages 175-200

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