Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. (2018). Pointfree topology version of image of real-valued continuous functions. Categories and General Algebraic Structures with Applications, 9(1), 59-75.
Abolghasem Karimi Feizabadi; Ali Akbar Estaji; Maryam Robat Sarpoushi. "Pointfree topology version of image of real-valued continuous functions". Categories and General Algebraic Structures with Applications, 9, 1, 2018, 59-75.
Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. (2018). 'Pointfree topology version of image of real-valued continuous functions', Categories and General Algebraic Structures with Applications, 9(1), pp. 59-75.
Karimi Feizabadi, A., Estaji, A., Robat Sarpoushi, M. Pointfree topology version of image of real-valued continuous functions. Categories and General Algebraic Structures with Applications, 2018; 9(1): 59-75.
Pointfree topology version of image of real-valued continuous functions
1Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
2Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
3Faculty of Mathematics and Computer Sciences,Hakim Sabzevari University, Sabzevar, Iran.
Abstract
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 $f:X\rightarrow\mathbb R$ and $ S \subseteq \mathbb R$.
1] Ball, R.N. and Hager, A.W., On the localic Yoshida representation of an archimedean lattice ordered group with weak order unit, J. Pure Appl. Algebra, 70 (1991), 17-43. [2] Ball, R.N. and Walters-Wayland, J., C- and C^* quotients on pointfree topology, Dissertations Mathematicae (Rozprawy Mat), 412 Warszawa (2002), 62 pp. [3] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures, (Curacao 1995), Kluwer Acad. Publ. (1997), 123-148. [4] Banaschewski, B., The real numbers in pointfree topology, Textos Mat. Sér. B 12, University of Coimbra, 1997. [5] Banaschewski, B. and Gilmour, C.R.A., Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 577-587. [6] Bhattacharjee, P., Knox, M.L., and McGovern,W.W., The classical ring of quotients of Cc(X), Appl. Gen. Topol. 15(2) (2014), 147-154. [7] Dube, T. and Ighedo, O., On z-ideals of pointfree function rings, Bull. Iran. Math. Soc. 40 (2014), 657-675. [8] Dube, T., Iliadis, S., Van Mill, J., and Naidoo, I., A Pseudocompact completely regular frame which is not spatial, Order 31(1) (2014), 115-120. [9] Ebrahimi, M.M. and Karimi Feizabadi, A., Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299. [10] Ebrahimi, M.M. and Mahmoudi, M., “Frames”, Technical Report, Department of Mathematics, Shahid Beheshti University, 1996. [11] Estaji, A.A., Karimi Feizabadi, A., and Abedi, M., Zero sets in pointfree topology and strongly z-ideals, Bull. Iran. Math. Soc 41(5) (2015), 1071-1084. [12] Estaji, A.A., Karimi Feizabadi, A., and Robat Sarpoushi, M., zc-Ideals and prime ideals in the ring RcL, to appear in Filomat. [13] Ghadermazi, M., Karamzadeh, O.A.S., and Namdari, M., On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69. [14] Gillman, L. and Jerison, M., “Rings of continuous functions”, Springer-Verlag, 1976. [15] Johnstone, P.T., “Stone spaces”, Cambridge University Press, 1982. [16] Karamzadeh, O.A.S., Namdari, M., and Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16 (2015), 183-207. [17] Karamzadeh, O.A.S. and Rostami, M., On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184. [18] Namdari, M. and Veisi, A., Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571. [19] Picado, J. and Pultr, A., “Frames and Locales: topology without points”, Birkhäuser/Springer, Basel AG, 2012.
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