The grade of zero
The grade of zero
◼ Strictly speaking, each exterior linear space of a Grassmann algebra will have its own zero, with a grade equal to the grade of the space. For simplicity, in GrassmannAlgebra, we choose to represent all these zeros by the one symbol 0. Thus, the symbol 0 has an ambiguous grade. To represent this ambiguous grade, GrassmannAlgebra uses the symbol ★0, called the grade of zero.
◼ A familiar case is the distinction between the zero vector of a vector space, and the zero scalar of its underlying field. These two zeros are clearly not identical.
◼ In normal circumstances, ★0 appears only in the output from the function or applied to a Grassmann expression. Its inclusion in the output from indicates that some element or elements in the expression are in fact zero due to their grades being outside the range of grades valid for the dimension of the declared space. Its inclusion in the output from indicates that has assessed the grade of some element or elements as negative.You will not normally use ★0 in input expressions.
◼ The grade of 0 is treated as of neither even nor odd grade.
Examples
Examples
◼ If the dimension of the space is 3, all expressions of grade greater than 3 are zero; hence their is returned as ★0.
★A;Grade1,x,x⋀y,x⋀y⋀z,,0,
x
4
z
{0,3,5,a}
{0,1,2,3,★0,★0,{0,3,a,★0}}
◼ computes a grade as if the dimension of the space were arbitrarily large, so the only element returning ★0 is the 0.
 | RawGrade |
x
4
z
{0,3,5,a}
{0,1,2,3,4,★0,{0,3,5,a}}
◼ The grade of 0 is neither even nor odd.
[0],
[0]
 | EvenGradeQ |
| OddGradeQ |
{False,False}
◼ The grade of 0 is the indeterminate scalar ★0. But the grade of this scalar is 0!
In[134]:=
Grade[0],Grade
| ★0 |
Out[134]=
{★0,0}
 |
|
""