Themba Dube - Department of Mathematical Sciences, University of South Africa, P.O. Box ۳۹۲, ۰۰۰۳ Unisa, South Africa.
Inderasan Naidoo - Department of Mathematical Sciences, University of South Africa, P.O. Box ۳۹۲, ۰۰۰۳ 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.

frame, remainder preservation, Stone-v{Cech} compactification, regular Lindel {o}f coreflection, realcompact coreflection, proper map, lax proper map