@article {
author = {Ighedo, Oghenetega and Kivunga, Grace Wakesho and Stephen, Dorca Nyamusi},
title = {A little more on ideals associated with sublocales},
journal = {Categories and General Algebraic Structures with Applications},
volume = {20},
number = {1},
pages = {175-200},
year = {2024},
publisher = {Shahid Beheshti University},
issn = {2345-5853},
eissn = {2345-5861},
doi = {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 $\mathcal RL$.},
keywords = {frame,locale,sublocale,pointfree function ring,Lindel\"{o}f,realcompact,paracompact},
url = {https://cgasa.sbu.ac.ir/article_104102.html},
eprint = {https://cgasa.sbu.ac.ir/article_104102_2ad4eee6ea89fcd4ab8026764c4c6e92.pdf}
}