Lattice Isomorphisms and Semilinear Transformations

Abstract

If we let ℒ and ℒ’ represent the geometries of two vector spaces V and V’ (over the same division ring), the maps of ℒ onto ℒ’ which preserve the order are natural objects of study from the geometric point of view. To every linear isomorphism of V onto V’ we can obviously associate an isomorphism of ℒ onto ℒ’. The question naturally arises whether every lattice isomorphism of ℒ onto ℒ can be constructed in this manner. It turns out that this is not quite so and that we have to include the so-called semilinear transformations in order to be able to describe all the lattice isomorphisms. At the same time, if we use the appropriate definitions, the class of semilinear transformations is adequate to describe all isomorphisms of ℒ onto ℒ’, even when V and V’ are defined over different but isomorphic division rings. Theorem 3.1, the fundamental result of this chapter, makes precise the remarks just made.