Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58534120160201The ring of real-continuous functions on a topoframe759413184ENAli AkbarEstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.AbolghasemKarimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.MohammadZarghaniMathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.Journal Article20150907 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 $alphainmathcal{R}tau $ is called $L$-{it extendable} real continuous function if and only if for every $rin mathbb{R}$, $bigvee^{L}_{rin mathbb R} (alpha(-,r)veealpha(r,-))'=top.$ 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 $.http://cgasa.sbu.ac.ir/article_13184_2f80da4a155068ca432d536a7217a6ab.pdf