A pointfree version of remainder preservation

Document Type: Research Paper


Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa.


Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambda$-proper or $upsilon$-proper in case the lifted homomorphism $h^{beta}colonbeta Ltobeta M$, $h^{lambda}colonlambda Ltolambda M$ or $h^{upsilon}colonupsilon Ltoupsilon M$ takes remainder to remainder. These turn out to be weaker forms of properness. Indeed, every proper homomorphism is $beta$-proper, every $beta$-proper homomorphism is $lambda$-proper, and $lambda$-properness is equivalent to $upsilon$-properness. A characterization of $beta$-proper maps in terms of pointfree rings of continuous functions is that they are precisely those whose induced ring homomorphisms contract free maximal ideals to free prime ideals.


[1] R.N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Dissertationes Mathematicae (Rozprawy Mat.), Vol. 412, (2002), 62pp.
[2] Banaschewski, B.: The real numbers in pointfree topology. Textos de Matematica Serie B, No. 12, Departamento de Matematica da Universidade de Coimbra,(1997).

[3] B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolinae 37(3) (1996), 577 - 587.
[4] B. Banaschewski and C. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categor. Struct. 9 (2001), 395 - 417.
[5] B. Banaschewski and C. Mulvey, Stone-  Cech compacti cation of locales. I, Houston J. Math. 6 (1980), 301 - 312.
[6] X. Chen, Stably closed frame homomorphisms, Cah. de Top. Geom. Di . Cat. 37(2) (1996), 123 - 144.
[7] T. Dube, Some characterizations of F-frames, Algebra Universalis 62(2009), 273 - 288.
[8] T. Dube, Some ring-theoretic properties of almost P-frames, Algebra Universalis 60 (2009), 145 - 162.
[9] T. Dube and I. Naidoo, On openness and surjectivity of lifted frame homomorphisms, Top. and its Appl. 157 (2010), 2159 - 2171.
[10] T. Dube and I. Naidoo, Erratum to [9], Top. Appl. 158 (2011), 2257-2259.
[11] R. Engelking, General Topology, Sigma Series in Pure Math., Berlin, 1989.
[12] J. Gutierrez Garcia, T. Kubiak and J. Picado, Localic real-valued functions: A general setting, J. Pure Appl. Algebra 213 (2009), 1064 - 1074.
[13] W. He and M. Luo, A note on proper maps of locales, Appl. Categor. Struct. (DOI 10.1007/s10485-009-9196-1).
[14] P.T. Johnstone, Stone Spaces, Cambridge Univ. Press, Cambridge, 1982.
[15] M. Korostenski and C.C.A. Labuschagne, Lax proper maps of locales, J. Pure Appl. Algebra 208 (2007), 655 - 664.
[16] J.J. Madden, -frames, J. Pure Appl. Algebra 70(1991), 107 - 127.
[17] J. Madden and J. Vermeer J, Lindelof locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99(1986), 473 - 480.
[18] N. Marcus, Realcompacti cation of frames, Comment. Math. Univ. Carolinae, 36(2)(1995), 349 - 358.
[19] J. Picado, A. Pultr and A. Tozzi, Locales In: Categorical foundations, 49-101, Encyclopedia Math. Appl., 97, Cambridge Univ. Press, Cambridge, 2004.
[20] J.J.C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92(1994), 79 -107.