The ring of real-continuous functions on a topoframe

Document Type : Research Paper


1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

2 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

3 Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.


 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 $.


[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),
[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,