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 ⊆ R. An α ∈ C(L) is said to be an overlap of S, denoted by α J S, whenever u ∩ S ⊆ 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) = ⋂{S ⊆ 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(α ∨ β) ⊆ pim(α) ∨ pim(β), pim(α ∧ β) ⊆ pim(α) ∧ pim(β), pim(αβ) ⊆ pim(α)pim(β) and pim(α+ β) ⊆ pim(α) + pim(β). * Corresponding author