When lifted frame homomorphisms are closed

It is known that any frame homomorphism into a compact regular frame is closed. So the lift h β : β L → β M of any frame homomorphism h : L → M to the Stone–Čech compactifications is closed regardless of whether h is closed or not. Here we study conditions which ensure that the lifts h λ : λ L → λ M and h υ : υ L → υ M to the Lindelöf and the realcompact coreflections, respectively, are closed.