Extensions of Galois Connections

Galois connections play a very important role in the theory of continuous lattices and their various generalizations. (See, for example, [1], [2], [a], [4], [5], [7] and [9].) Morphisms of continuous lattices, as defined in [2], are precisely those upper adjoints of Galois connections which preserve directed sups. In [1] Bandelf and Ernd suggested that the right choice of morphisms for Z-continuous posets are the upper adjoints of Galois connections which preserve Z-sups. Results in [9] further justify this choice. So, it is important to find conditions under which Galois connections between the bases of Z-continuous posers can be extended to Galois connections between the posers themselves. This investigation is all the more interesting, since a result of Wright, Wagner, and Thatcher [10] implies that for a union complete Z, a monotone function from a basis B of a Z-continuous poser P can be extended to a Z-continuous map from P if and only if P is a Z-algebraic poser and B is the set of all Z-compact elements. Recall that a subset system Z is a function which assigns to each poser P a set Z(P) of subsets of P such that (i) for all P, all singletons of P are in Z(P); (ii) if f' P -• Q is a monotone function and S is in Z(P), then f(S) is in Z(Q). NOTATION' Let P be any poser and let S C_ P.