Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58531120131201A pointfree version of remainder preservation27584264ENThemba DubeDepartment of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South
Africa.Inderasan NaidooDepartment of Mathematical Sciences, University of South
Africa, P.O. Box 392, 0003 Unisa, South Africa.Journal Article20140118 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.http://cgasa.sbu.ac.ir/article_4264_91ce60eb77415d9197885588177906a7.pdf