A little more on ideals associated with sublocales

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