Inverses of disjointness preserving operators

A linear operator T : X --> Y between vector lattices is said to be disjointness preserving if T sends disjoint elements in X to disjoint elements in Y. The bijective disjointness preserving operators are the central object of this work. The following three types of results are obtained. 1) For a bijective disjointness preserving operator T : X --> Y a number of results are proved demonstrating that (under some mild additional conditions on the vector lattices) the inverse operator T-1 is also disjointness preserving, and furthermore the vector lattices X and Y are order isomorphic. Moreover, for a Dedekind complete vector lattice X necessary and sufficient conditions are found under which any bijective disjointness preserving operator on X has the disjointness preserving inverse. We prove also that any d-isomorphic Dedekind complete vector lattices are order isomorphic. 2) A general method is presented for producing bijective disjointness preserving operators T : X --> Y (with various conditions on X and Y) for which T-1 fails to preserve disjointness. 3) A general method is presented for producing bijective operators T : X --> Y for which both T and T-1 preserve disjointness but X and Y are not order isomorphic. It should be pointed out that the results referred to in 1)-3) answer several well known open problems concerning operators on vector lattices.