The ring of real-continuous functions on a topoframe | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 6، دوره 4، شماره 1، اردیبهشت 2016، صفحه 75-94 اصل مقاله (511.66 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| Ali Akbar Estaji1؛ Abolghasem Karimi Feizabadi2؛ Mohammad Zarghani3 | ||
| 1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. | ||
| 2Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran. | ||
| 3Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. | ||
| چکیده | ||
| 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 $\alpha\in\mathcal{R}\tau $ is called $L$-{\it extendable} real continuous function if and only if for every $r\in \mathbb{R}$, $\bigvee^{L}_{r\in \mathbb R} (\alpha(-,r)\vee\alpha(r,-))'=\top.$ 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 $. | ||
| کلیدواژهها | ||
| frame؛ Topoframe؛ Ring of real continuous functions؛ Archimedean ring؛ $f$-ring | ||
| مراجع | ||
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[1] R.N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Disser- tationes Math. (Rozprawy Mat.) 412 (2002), 1-61. [2] R.N. Ball and A.W. Hager, On the localic Yosida representation of an archimedean lattice ordered group with weak unit, J. Pure Appl. Algebra 70 (1991), 17-43. [3] B. Banaschewski, On the function ring functor in pointfree topology, Appl. Categ. Structures 13 (2005), 305-328. [4] B. Banaschewski, The real numbers in pointfree topology", Textos de Mathematica (Series B) 12, University of Coimbra, 1997. [5] T. Dube, A note on the socle of certain types of f-rings, Bull. Iranian Math. Soc. 38(2) (2012), 517-528. [6] T. Dube, Extending and contracting maximal ideals in the function rings of pointfree topology, Bull. Math. Soc. Sci. Math. Roumanie 55(103)(4) (2012), 365-374. [7] T. Dube, Some algebraic characterizations of F-frames, Algebra Universalis 62 (2009), 273-288. [8] T. Dube, Some ring-theoretic properties of almost P-frames, Algebra Universalis 60 (2009), 145-162. [9] M.M. Ebrahimi and A. Karimi Feizabadi, Prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299. [10] A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Strongly xed ideals in C(L) and compact frames, Archivum Mathematicum(Brno), Tomus 51 (2015), 1-12. [11] A.A. Estaji, A. Karimi Feizabadi, and M. Abedi, Zero set in pointfree topology and strongly z-ideals, Bull. Iranian Math. Soc. 41(5) (2015), 1071-1084. [12] M. J. Ferreira, J. Gutierrez Garca, J. Picado, Completely normal frames and real- valued functions, Topology Appl. 156 (2009), 2932-2941. [13] L. Gillman and M. Jerison, Rings of continuous functions", Springer-Verlag, 1976. [14] C.R.A. Gilmour, Realcompact spaces and regular -frames, Math. Proc. Camb. Phil. Soc. 96 (1984) 73-79. [15] A. Karimi Feizabadi, A.A. Estaji, and M. Zarghani, The ring of real-valued functions on a frame, Preprint. [16] P.T. Johnstone, Stone Spaces", Cambridge University Press, Cambridge, 1982. [17] J. Picado and A. Pultr, Frames and Locales: topology without points", Frontiers in Mathematics, Springer, Basel 2012. [18] M. Zarghani, A.A. Estaji, and A. Karimi Feizabadi, Modi ed pointfree topology, Preprint. | ||
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