Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories image of ω-complete posets, image of directed complete posets and image of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of image) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in image (image, image) exponentiable in image w.r.t. the Scott topology is exponentiable also in image (image, image). We prove that the converse is true in the category image, but neither in image, nor in image.