%0 Journal Article %T The ring of real-continuous functions on a topoframe %J Categories and General Algebraic Structures with Applications %I Shahid Beheshti University %Z 2345-5853 %A Estaji, Ali Akbar %A Karimi Feizabadi, Abolghasem %A Zarghani, Mohammad %D 2016 %\ 02/01/2016 %V 4 %N 1 %P 75-94 %! The ring of real-continuous functions on a topoframe %K frame %K Topoframe %K Ring of real continuous functions %K Archimedean ring %K $f$-ring %R %X  A topoframe, denoted by $L_{ \tau}$,  is a pair $(L, \tau)$ consisting of a frame $L$ and a subframe $ \tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $\tau $-real-continuous function on a frame $L$ and the set of real continuous functions $\mathcal{R}L_\tau $ as an $f$-ring. We show that $\mathcal{R}L_{ \tau}$ 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 $\mathcal{R}L_{ \tau}$ is isomorphic to a sub-$f$-ring of $\mathcal{R}\tau .$ Let ${\tau}$ be a topoframe on a frame $L$. The frame map $\alpha\in\mathcal{R}\tau $ is called $L$-{\it extendable} real continuous function if and only if for every $r\in \mathbb{R}$, $\bigvee^{L}_{r\in \mathbb R} (\alpha(-,r)\vee\alpha(r,-))'=\top.$ Finally, we prove that $\mathcal{R}^{L}{\tau}\cong \mathcal{R}L_{\tau}$ as $f$-rings, where $\mathcal{R}^{L}{\tau}$ is the set all of $L$-extendable real continuous functions of $ \mathcal{R}\tau $. %U https://cgasa.sbu.ac.ir/article_13184_2f80da4a155068ca432d536a7217a6ab.pdf