  The theory of Nambu-Poisson structures on manifolds is extended to the
context of Lie algebroids, in a natural way based on the Vinogradov bracket
associated with Lie algebroid cohomology. We show that, under certain
assumptions, any Nambu-Poisson structure on a Lie algebroid is
decomposable.Also, we introduce the concept of a higher order Dirac structure
on a Lie algebroid. This allows to describe both Nambu-Poisson structures on
Lie algebroids and Dirac structures on manifolds in the same setting.
