  Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and
Batalin-Vilkovisky algebras are explored. These are defined in terms of various
antisymmetric bilinear operations satisfying weakened versions of the Jacobi
identity, as well as in terms of operations involving more than two variables
of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra-the
algebra of alternating forms on a vector space so that Lie brackets correspond
to square zero derivations of this algebra-and multialgebra generalizations
thereof. The higher homotopies are phrased in terms of these multialgebras.
Applications to foliations are discussed: objects which serve as replacements
for the Lie algebra of vector fields on the "space of leaves" and for the
algebra of multivector fields are developed, and the spectral sequence of a
foliation is shown to arise as a special case of a more general spectral
sequence including as well the Hodge-de Rham spectral sequence.
