Exponentiable morphisms of domains

Given a map f in the category w-Cpo of w-complete posets, exponentiability of f in w-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in w-Cpo, showing the existence of partial products of the two-point ordered set S = {0 < 1} (Theorem 1.8). Using this characterization and the embedding via the Scott topology of w-Cpo in the category Top of topological spaces, we can compare exponentiability in each setting, obtaining that a morphism in w-Cpo, exponentiable both in Top and in Pos, is exponentiable also in w-Cpo. Furthermore we show that the exponentiability in Top and in Pos are independent from each other.