let $l$ be a completely regular frame and $mathcal{r}l$ be the ringof continuous real-valued functions on $l$. we study the frame$mathfrak{o}(min(mathcal{r}l))$ of minimal prime ideals of$mathcal{r}l$ in relation to $beta l$. for $iinbeta l$, denoteby $textit{textbf{o}}^i$ the ideal${alphainmathcal{r}lmidcozalphain i}$ of $mathcal{r}l$. weshow that sending $i$ to the set of minimal prime ideals...

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 continu...

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 t...

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 continu...

 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 generali...

‎For any archimedean$f$-ring $A$ with unit in whichbreak$awedge‎ ‎(1-a)leq 0$ for all $ain A$‎, ‎the following are shown to be‎ ‎equivalent‎: ‎ ‎1‎. ‎$A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all‎ ‎integer-valued continuous functions on some frame $L$‎. 2‎. ‎$A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$‎ ‎of all integer-valued continuous functions‎, ‎in the usual se...

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 realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...

this paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

‎for any archimedean$f$-ring $a$ with unit in whichbreak$awedge‎‎(1-a)leq 0$ for all $ain a$‎, ‎the following are shown to be‎‎equivalent‎:‎‎1‎. ‎$a$ is isomorphic to the $l$-ring ${mathfrak z}l$ of all‎‎integer-valued continuous functions on some frame $l$‎. 2‎. ‎$a$ is a homomorphic image of the $l$-ring $c_{bbb z}(x)$‎‎of all integer-valued continuous functions‎, ‎in the usual sense‎,‎on som...