%0 Journal Article
%T A pointfree version of remainder preservation
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Dube, Themba
%A Naidoo, Inderasan
%D 2013
%\ 12/01/2013
%V 1
%N 1
%P 27-58
%! A pointfree version of remainder preservation
%K frame
%K remainder preservation
%K Stone-v{Cech} compactification
%K regular Lindel"{o}f coreflection
%K realcompact coreflection
%K proper map
%K lax proper map
%R
%X 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.
%U https://cgasa.sbu.ac.ir/article_4264_91ce60eb77415d9197885588177906a7.pdf