  We build generalizations of the Grassmann algebras from a few simple
assumptions which are that they are graded, maximally symmetric and contain an
ordinary Grassmann algebra as a subalgebra. These algebras are graded by
Z_{n}^{3} and display surprising properties that indicate their possible
application to the modelization of quark fields. We build the generalized
supersymmetry generators based on these algebras and their derivation
operators. These generators are cubic roots of the usual supersymmetry
generators.
