Let and be vector lattices. A linear map is disjointness preserving if whenever in ; if is bijective and is also disjointness-preserving then is said to be a -isomorphism. The main results of this monograph include:
(A) a characterization of those Dedekind-complete lattices having the property that every disjointness-preserving bijection with domain is a -isomorphism, and
(B) a theorem to the effect that for Dedekind-complete vector lattices, -isomorphism implies order isomorphism.
Partial results along these lines were announced in [the authors, “Functional analysis and economic theory”. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, Berlin: Springer, 1-8 (1998; Zbl 0916.47026)]. Besides being more definitive and complete, the present account also initiates a new perspective: rather than regarding disjointness preserving bijections which fail to be -isomorphisms as aberrant curiosities, their existence is now taken as an opportunity to learn about the structures of domain and range spaces. This new perspective is reflected in the rich variety of counterexamples presented and in the full treatments of concepts like ‘determining/cofinal families of band projections’, ‘-splitting numbers’, ‘essentially constant functions’, and ‘cofinal universal completeness’ leading up to (A).