A pointfree version of remainder preservation

Recall that a continuous function f: x→ y between tychono spaces is proper if and only if the stone extension f^β : βx→ βy takes remainder to remainder, in the sense that f^β[βx-x] ⫃ βy-y . we introduce the notion of taking remainder to remainder to frames, and, using it, we defne a frame homomorphism h: l→ m to be β-proper, λ-proper or -proper in case the lifted homomorphism h^β: βl →βm, h : λl → λm or h: vl→ vm takes remainder to remainder. these turn out to be weaker forms of properness. indeed, every proper homomorphism is β-proper, every β-proper homomorphism is v-proper, and λ- properness is equivalent to -properness. a characterization of λ-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.