 Graded functions on such a chart are formula_27-valued functions

formula_39,

where formula_40 are smooth real functions on formula_41 and formula_42 are odd generating elements of the Grassmann algebra formula_27.Given a graded manifold formula_2, graded derivations of the structure ring of graded functions formula_17 are called graded vector fields on formula_2.They constitute a real Lie superalgebra formula_47 with respect to the superbracket

formula_48,

where formula_49 denotes the Grassmann parity of formula_50.