The ring of real-valued functions on a frame

Document Type: Research Paper


1 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.


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)$.


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