TY - JOUR
ID - 104102
TI - A little more on ideals associated with sublocales
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Ighedo, Oghenetega
AU - Kivunga, Grace Wakesho
AU - Stephen, Dorca Nyamusi
AD - Department of Mathematics, Chapman University, P.O. Box 92866, California, U.S.A.
AD - Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa.
AD - Deparment of Mathematics and Physics, Technical University of Mombasa, P.O. Box 90420-80100, Mombasa, Kenya.
Y1 - 2024
PY - 2024
VL - 20
IS - 1
SP - 175
EP - 200
KW - frame
KW - locale
KW - sublocale
KW - pointfree function ring
KW - Lindel\"{o}f
KW - realcompact
KW - paracompact
DO - 10.48308/cgasa.2023.234093.1456
N2 - 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 $\mathcal RL$.
UR - https://cgasa.sbu.ac.ir/article_104102.html
L1 - https://cgasa.sbu.ac.ir/article_104102_2ad4eee6ea89fcd4ab8026764c4c6e92.pdf
ER -