Slimming and regularization of cozero maps

Document Type : Research Paper


1 Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.

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


Cozero maps are generalized forms of cozero elements. Two particular cases of cozero maps, slim and regular cozero maps, are significant. In this paper we present methods to construct slim and regular cozero maps from a given  cozero map. The construction of the slim and the regular cozero map from a cozero map are called slimming and regularization of the cozero map, respectively. Also, we prove that the slimming and regularization create reflector functors, and so we may say that they are the best method of constructing slim and regular cozero maps, in the sense of category theory.
 Finally, we give slim regularization for a cozero map $c:M\rightarrow L$ in the general case where $A$ is not a ${\Bbb Q}$-algebra. We use the ring and module of fractions, in this construction process.


[1] Adamek, J., H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories", John Wiley and Sons, Inc., 1990.
[2] Ball, R.N. and A. Pultr, Quotients and colimits of kappa-quantales, Topology Appl. 158 (2011), 2294-2306.
[3] Banaschewski, B., Propositional logic, frames, and fuzzy algebra, Quaest. Math. 22 (1999), 481-508.
[4] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures (Curaccao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123-148.
[5] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.
[6] Banaschewski, B. and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ.  Carolin. 37(3) (1996), 577-587.
[7] Banaschewski, B. and C. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9(4) (2001), 395-417.
[8] Banaschewski, B. and C. Gilmour, Cozero bases of frames, J. Pure Appl. Algebra 157(1) (2001), 1-22.
[9] Dube, T., Concerning the frame of minimal prime ideals of pointfree function rings, Categ. Gen. Algebr. Struct. Appl. 1(1) (2013), 11-26.
[10] Ebrahimi, M.M. and A. Karimi Feizabadi, Spectra of `-Modules, J. Pure Appl. Algebra 208 (2007), 53-60.
[11] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree version of Kakutani duality, Order, 22 (2005), 241-256.
[12] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299.
[13] Ebrahimi, M.M., A. Karimi, and M. Mahmoudi, Pointfree spectra of Riesz spaces, Appl. Categ. Structures 12 (2004), 397-409.
[14] Gillman, L. and M. Jerison, Rings of Continuous Function", Graduate Texts in Mathematics 43, Springer-Verlag, 1979.
[15] Gutierrez Garccia, J., J. Picado, and A. Pultr, Notes on point-free real functions and sublocales, Categorical Methods in Algebra and Topology, Textos de Matematica, 46, University of Coimbra, (2014), 167-200.
[16] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383-394.
[17] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.
[18] Matutu, P., The cozero part of a biframe, Kyungpook Math. J., 42(2) (2002), 285-295.