Concerning some variants of C-embedding in pointfree topology

We study three types of quotient maps of frames which are closely related to C- and C⁎-quotient maps. We call them C1-, strong C1-, and uplifting quotient maps. C1-quotient maps are precisely those whose induced ring homomorphisms contract maximal ideals to maximal ideals. We show that every homomorphism onto a frame is a C1-, a strong C1-, or an uplifting quotient map iff the frame is pseudocompact, compact, or almost compact and normal, respectively. These quotient maps are used to characterize normality and also certain weaker forms of normality in a manner akin to the characterization of normal frames as those for which every closed quotient map is a C-quotient map. Under certain conditions, we show that the Stone extension of a quotient map is C1-, strongly C1- or uplifting if the map has the corresponding property.