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

Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $Ssubseteq{mathbb R}$. An $alphain C(L)$ is said to be an overlap of $S$, denoted by $alphablacktriangleleft S$, whenever $ucap Ssubseteq vcap S$ implies $alpha(u)leqalpha(v)$ 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 ${rm pim}(alpha)=bigcap{Ssubseteq{mathbb R}:~alphablacktriangleleft S}$, 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 ${rm pim}(alphaveebeta)subseteq {rm pim}(alpha)cup {rm pim}(beta)$, ${rm pim}(alphawedgebeta)subseteq {rm pim}(alpha)cap {rm pim}(beta)$, ${rm pim}(alphabeta)subseteq {rm pim}(alpha){rm pim}(beta)$ and ${rm pim}(alpha+beta)subseteq {rm pim}(alpha)+{rm pim}(beta)$.