The ring of real-valued functions on a frame

In this paper, we define and study the notion of the real-valued functions on a frame L. We show that F(L) , consisting of all frame homomorphisms from the power set of \mathbb{R} to a frame L, is an f-ring, as a generalization of all functions from a set X into \mathbb R. Also, we show that F(L) is isomorphic to a sub-f-ring of \mathcal{R}(L), the ring of real-valued continuous functions on L. Furthermore, for every frame L, there exists a Boolean frame B such that F(L) is a sub-f-ring of F(B).