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کاربرد نوع شرط:
- جایگاه : پژوهشی
- مجله: Categories and General Algebraic Structures with Applications
- نوع مقاله: Journal Article
- کلمات کلیدی: frame,$f$-ring,ring of real-valued continuous functions,countable image
- چکیده:
- چکیده انگلیسی: 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$.- انتشار مقاله: 28-12-1395
- نویسندگان: Abolghasem Karimi Feizabadi,Ali Akbar Estaji,Maryam Robat Sarpoushi
- مشاهده
- جایگاه : پژوهشی
- مجله: Categories and General Algebraic Structures with Applications
- نوع مقاله: Journal Article
- کلمات کلیدی: Minimal ideal,Socle,real-valued functions ring,ring with property $(A)$
- چکیده:
- چکیده انگلیسی: For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms. Also it is shown that not only $ mathcal{F}_{mathcal P}L$ has Property (A) but also if $L$ has a finite number of atoms then the residue class ring $ mathcal{F}_{mathcal P}L/mathrm{Soc}( mathcal{F}_{mathcal P}L)$ has Property (A).
- انتشار مقاله: 22-09-1395
- نویسندگان: Ali Asghar Estaji,Ebrahim Hashemi,Ali Akbar Estaji
- مشاهده
- جایگاه : پژوهشی
- مجله: Categories and General Algebraic Structures with Applications
- نوع مقاله: Journal Article
- کلمات کلیدی: frame,$f$-ring,Ring of real-valued functions
- چکیده:
- چکیده انگلیسی: 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)$.
- انتشار مقاله: 06-10-1394
- نویسندگان: Abolghasem Karimi Feizabadi,Ali Akbar Estaji,Mohammad Zarghani
- مشاهده
- جایگاه : پژوهشی
- مجله: Algebraic Structures and Their Applications
- نوع مقاله: Journal Article
- کلمات کلیدی: Lattice-ordered ring,Zero-dimensional frame,$F_{\mathcal P}$-realcompact,Real Riesz map,Free ideal,Real ideal
- چکیده:
- چکیده انگلیسی: Let $mathcal F_{mathcal P}( L)$ ($mathcal F_{mathcal P}^{*}( L)$) be the $f$-rings of all (bounded) frame maps from $mathcal P(mathbb R)$ to a frame $L$. $mathcal F_{{mathcal P}_{infty}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ such that ${uparrow}f(-frac 1n, frac 1n)$ is compact for any $ninmathbb N$ and the subring $mathcal F_{{mathcal P}_{K}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ such that ${{,mathrm{coz},}}(f)$ is compact. We introduce and study the concept of real ideals in $mathcal F_{mathcal P}( L)$ and $mathcal F_{mathcal P}^*( L)$. We show that every maximal ideal of $mathcal F_{mathcal P}^{*}( L)$ is real, and also we study the relation between the conditions ``$L$ is compact" and ``every maximal ideal of $mathcal F_{mathcal P}(L)$ is real''. We prove that for every nonzero real Riesz map $varphi colon mathcal F_{mathcal P}( L)rightarrow mathbb R$, there is an element $p$ in $Sigma L$ such that $varphi=widetilde {p_{{{,mathrm{coz},}}}}$
if $L$ is a zero-dimensional frame for which $B(L)$ is a sub-$sigma$-frame of $L$ and every maximal ideal of $mathcal F_{mathcal P}( L)$ is real. We show that $mathcal F_{{mathcal P}_{infty}}(L)$ is equal to the intersection of all free maximal ideals of $ mathcal F_{mathcal P}^{*}(L) $ if $B(L)$ is a sub-$sigma$-frame of a zero-dimensional frame $L$ and also, $mathcal F_{{mathcal P}_{K}}(L)$ is equal to the intersection of all free ideals $mathcal F_{mathcal P}( L)$ (resp., $mathcal F_{mathcal P}^*( L)$) if $L$ is a zero-dimensional frame. Also, we study free ideals and fixed ideals of $mathcal F_{{mathcal P}_{infty}}( L)$ and $mathcal F_{{mathcal P}_{K}}( L)$.- انتشار مقاله: 04-01-1399
- نویسندگان: Ali Estaji,Ahmad Mahmoudi Darghadam
- مشاهده