The ring of real-valued functions on a frame

Document Type : Research Paper


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

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


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)$.


[1] R.N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Dissertationes
Math. (Rozprawy Mat.) 412 (2002), 1-61.
[2] B. Banaschewski, On the function rings of pointfree topology, Kyungpook Math. J.
48 (2008), 195-206.
[3] B. Banaschewski, Pointfree topology and the spectra of f-rings, Ordered algebraic
structures, (Curaçoa, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123-148.
[4] B. Banaschewski, Ring theory and pointfree topology, Topology Appl. 137 (2004),
[5] B. Banaschewski, “The real numbers in pointfree topology", Textos de Mathematica
(Series B), Vol. 12, University of Coimbra, 1997.
[6] T. Dube and O. Ighedo, On z-ideals of pointfree function rings, Bull. Iranian Math.
Soc. 40(3) (2014), 657-675.
[7] T. Dube, A note on the socle of certain types of f-rings, Bull. Iranian Math. Soc.
38(2) (2012), 517-528.
[8] T. Dube, Contracting the socle in 1rings of continuous functions, Rend. Semin. Mat.
Univ. Padova 123 (2010), 37-53.
[9] A.A. Estaji, A. Karimi Feizabadi and M. Zarghani, The ring of real-continuous
functions on a topoframe, Categ. General Alg. Structures Appl. 4 (2016), 75-94
[10] L. Gillman and M. Jerison, “Rings of Continuous Functions", Springer-Verlag, 1976.