Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection such that if and only if for all . We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to , and for Banach lattices that contain a closed sublattice order isomorphic to .