Pre-image of functions in $C(L)$

Abstract

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 ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$.