Karimi Feizabadi, A., Estaji, A., Zarghani, M. (2016). The ring of real-valued functions on a frame. Categories and General Algebraic Structures with Applications, 5(1), 85-102.

Abolghasem Karimi Feizabadi; Ali Akbar Estaji; Mohammad Zarghani. "The ring of real-valued functions on a frame". Categories and General Algebraic Structures with Applications, 5, 1, 2016, 85-102.

Karimi Feizabadi, A., Estaji, A., Zarghani, M. (2016). 'The ring of real-valued functions on a frame', Categories and General Algebraic Structures with Applications, 5(1), pp. 85-102.

Karimi Feizabadi, A., Estaji, A., Zarghani, M. The ring of real-valued functions on a frame. Categories and General Algebraic Structures with Applications, 2016; 5(1): 85-102.

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

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

Abstract

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), 21–37. [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.