The ring of real-continuous functions on a topoframe

A topoframe, denoted by Lτ, is a pair (L,τ) consisting of a frame L and a subframe τ all of whose elements are complementary elements in L. In this paper, we define and study the notions of a τ-real-continuous function on a frame L and the set of real continuous functions RLτ as an f-ring. We show that RLτ is actually a generalization of the ring C(X) of all real-valued continuous functions on a completely regular Hausdorff space X. In addition, we show that RLτ is isomorphic to a sub-f-ring of Rτ. Let τ be a topoframe on a frame L. The frame map α∈Rτ is called L-{\it extendable} real continuous function if and only if for every r∈R, ⋁Lr∈R(α(−,r)∨α(r,−))′=⊤. Finally, we prove that RLτ≅RLτ as f-rings, where RLτ is the set all of L-extendable real continuous functions of Rτ.