Pre-image of functions in C(L)

Let C(L) be the ring of all continuous real functions on a frame L and S Ҫ R. An Ҫ 2 C(L) is said to be an overlap of S, denoted by Ҫ J S, whenever u / S Ҫ v / S implies α(u) 6 α(v) for every open sets u and v in R. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in 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 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 pim(α) = T {S Ҫ R : α J 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 pim(αβ) Ҫ pim(α) _ pim(β), pim(α ^ β) Ҫ pim(α) ^ pim(β), pim(αβ) Ҫ pim(α)pim(β) and pim(α + β) pim(α) + pim(β).