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 ff-ring. we show that rlτis 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 thatrlτ 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- 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 ff-rings, where rlτ is the set all of l-extendable real continuous functions of rτ.