  A deformed $q$-calculus is developed on the basis of an algebraic structure
involving graded brackets. A number operator and left and right shift operators
are constructed for this algebra, and the whole structure is related to the
algebra of a $q$-deformed boson. The limit of this algebra when $q$ is a $n$-th
root of unity is also studied in detail. By means of a chain rule expansion,
the left and right derivatives are identified with the charge $Q$ and covariant
derivative $D$ encountered in ordinary/fractional supersymmetry and this leads
to new results for these operators. A generalized Berezin integral and
fractional superspace measure arise as a natural part of our formalism. When
$q$ is a root of unity the algebra is found to have a non-trivial Hopf
structure, extending that associated with the anyonic line. One-dimensional
ordinary/fractional superspace is identified with the braided line when $q$ is
a root of unity, so that one-dimensional ordinary/fractional supersymmetry can
be viewed as invariance under translation along this line. In our construction
of fractional supersymmetry the $q$-deformed bosons play a role exactly
analogous to that of the fermions in the familiar supersymmetric case.
