Pre-image of functions in C(L)

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} (۲۰۱۸). 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).

Keywords:

frame , Pointfree topology , C(L) , pre-image , prime ideal and maximal ideal in frames , f-algebra