Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E ^* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e., mod(E)?=?0. Further, the relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as the modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and of a Poisson map.