%0 Journal Article
%T Pointfree topology version of image of real-valued continuous functions
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Karimi Feizabadi, Abolghasem
%A Estaji, Ali Akbar
%A Robat Sarpoushi, Maryam
%D 2018
%\ 07/01/2018
%V 9
%N 1
%P 59-75
%! Pointfree topology version of image of real-valued continuous functions
%K frame
%K ring of real-valued continuous functions
%K countable image
%K $f$-ring
%R 10.29252/cgasa.9.1.59
%X Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree version of $C_c(X).$The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {mathcal{R}} L$ into the power set of $mathbb R$, namely overlap . Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $mathop{hbox{Im}} (f) subseteq S$ for every continuous function $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.
%U https://cgasa.sbu.ac.ir/article_50745_d90d55e08316779860740922b0388294.pdf