Exponentiable morphisms of domains

Given a map f in the category ω-Cpo of ω-complete posets, exponentiability of f in ω-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We investigate the extra conditions needed on f exponentiable in Pos to be exponentiable in ω-Cpo by showing the existence of partial products of the two-point ordered set S={0<1} (Theorem 2.8). Using this characterisation and the embedding through the Scott topology of ω-Cpo in the category Top of topological spaces, we compare exponentiability in each setting and find that a morphism in ω-Cpo that is exponentiable in both Top and Pos is exponentiable in ω-Cpo also. Furthermore, we show that the exponentiability in Top and Pos are independent of each other.