  This paper investigates higher order generalizations of well known results
for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid
structures correspond to $n$-ary generalization of Gerstenhaber algebras and
are implied by $n$-ary generalization of linear Poisson structures on the dual
bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special
bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid
(of order $n$). It is further demonstrated that such bialgebroids canonically
induce a Nambu-Poisson structure on the base manifold. Finally, the tangent
space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid
over a point.
