Disjointly homogeneous Banach lattices: Duality and complementation

We study several properties of disjointly homogeneous Banach lattices with a special focus on two questions: the self-duality of this class and the existence of disjoint sequences spanning complemented subspaces. Various results around these problems are given. In particular, we provide examples of reflexive disjointly homogeneous spaces whose dual spaces are not disjointly homogeneous.