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.