  In the present article we investigate the possibility of combining the usual
Grassmann algebras with their ternary Z_3-graded counterpart, thus creating a
more general algebra with coexisting quadratic and cubic constitutive
relations.
  We recall the classification of ternary and cubic algebras according to the
symmetry properties of ternary products under the action of the S_3 permutation
group. Instead of only two kinds of binary algebras, symmetric or
antisymmetric, here we get four different generalizations of each of these two
cases. Then we study a particular case of algebras generated by two types of
variables, the generators of Grassmann algebra and generators of ternary analog
of Grassmann algebra, satisfying quadratic and cubic relations respectively,
i.e. the generators of Grassmann algebra anticommute and a triple product of
any three generators of a ternary analog of Grassmann algebra is equal to
cyclic permutation of generators in this product multiplied by a primitive 3rd
root of unity. The invariance group of the generalized algebra is introduced
and investigated.
