pre-image of functions in c(l)

Let c(l) be the ring of all continuous real functions on a frame l and s ⊆ r. an α ∈ c(l) is said to be an overlap of s, denoted by α ◄ s, whenever u ∩ s ⊆ v ∩ s implies α(u) ≤ 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) = ∩{s ⊆ r : α ◄ 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 (α v β) ⊆ pim(α) v pim(β), pim(α λ β) ⊆ pim(α) λ pim(β), pim(αβ) ⊆ pim(α)pim(β) and pim(α + β) ⊆ pim(α) + pim(β).