@article { author = {Karimi Feizabadi, Abolghasem and Estaji, Ali Akbar and Zarghani, Mohammad}, title = {The ring of real-valued functions on a frame}, journal = {Categories and General Algebraic Structures with Applications}, volume = {5}, number = {1}, pages = {85-102}, year = {2016}, publisher = {Shahid Beheshti University}, issn = {2345-5853}, eissn = {2345-5861}, doi = {}, abstract = {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)$.}, keywords = {frame,$f$-ring,Ring of real-valued functions}, url = {https://cgasa.sbu.ac.ir/article_14685.html}, eprint = {https://cgasa.sbu.ac.ir/article_14685_3fb4a5800764dc102f3cbb565968e45d.pdf} }