TY - JOUR
ID - 100691
TI - Pre-image of functions in $C(L)$
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Rezaei Aliabad, Ali
AU - Mahmoudi, Morad
AD - Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
AD - Department of of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Y1 - 2021
PY - 2021
VL - 15
IS - 1
SP - 35
EP - 58
KW - frame
KW - Pointfree topology
KW - $C(L)$
KW - pre-image
KW - prime ideal and maximal ideal in frames
KW - $f$-algebra
DO - 10.52547/cgasa.15.1.35
N2 - Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $S\subseteq{\mathbb R}$. An $\alpha\in C(L)$ is said to be an overlap of $S$, denoted by $\alpha\blacktriangleleft S$, whenever $u\cap S\subseteq v\cap S$ implies $\alpha(u)\leq\alpha(v)$ for every open sets $u$ and $v$ in $\mathbb{R}$. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in {\it Pointfree version of image of real-valued continuous functions} (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by ${\rm pim}$, as a pointfree version of the image of real-valued continuous functions on a topological space $X$. We investigate this concept and in addition to showing ${\rm pim}(\alpha)=\bigcap\{S\subseteq{\mathbb R}:~\alpha\blacktriangleleft S\}$, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have ${\rm pim}(\alpha\vee\beta)\subseteq {\rm pim}(\alpha)\cup {\rm pim}(\beta)$, ${\rm pim}(\alpha\wedge\beta)\subseteq {\rm pim}(\alpha)\cap {\rm pim}(\beta)$, ${\rm pim}(\alpha\beta)\subseteq {\rm pim}(\alpha){\rm pim}(\beta)$ and ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$.
UR - https://cgasa.sbu.ac.ir/article_100691.html
L1 - https://cgasa.sbu.ac.ir/article_100691_2f5fc0016cb257b218617881ce982e87.pdf
ER -