A pointfree version of remainder preservation

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.

Keywords:

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