Grassmann Algebra

TheRegressiveProduct.nb 9
Translating this into the terminology used in this book we have:
If q and r are the grades of two elements A and B, and n that of the underlying linear
space, then the grade of the product [A B] is first equal to q+r if q+r is smaller than n, and
second equal to q+rÐn if q+r is greater than or equal to n.
Translating this further into the notation used in this book we have:
�A B� � A p � B q
�A B� � A p � B q
p � q � n
p � q � n
Grassmann called the product [A B] in the first case a progressive product, and in the second
case a regressive product.
In the equivalence above, Grassmann has opted to define the product of two elements whose
grades sum to the dimension n of the space as regressive, and thus a scalar. However, the more
explicit notation that we have adopted identifies that some definition is still required for the
progressive (exterior) product of two such elements.
The advantage of denoting the two products differently enables us to correctly define the
exterior product of two elements whose grades sum to n, as an n-element. Grassmann by his
choice of notation has had to define it as a scalar. In modern terminology this is equivalent to
confusing scalars with pseudo-scalars. A separate notation for the two products thus avoids this
tensorially invalid confusion.
We can see how then, in not being explicitly denoted, the regressive product may have become
lost.
3.4 The Duality Principle
The dual of a dual
The duality of the axiom sets for the exterior and regressive products is completed by requiring
that the dual of a dual of an axiom is the axiom itself. The dual of a regressive product axiom
may be obtained by applying the following algorithm:
1. Replace � with �, and the grades of elements and spaces by their complementary grades.
2. Replace arbitrary grades m with nÐm', k with nÐk'. Drop the primes.
This differs from the algorithm for obtaining the dual of an exterior product axiom only in the
replacement of � with � instead of vice versa.
It is easy to see that applying this algorithm to the regressive product axioms generates the
original exterior product axiom set.
We can combine both transformation algorithms by restating them as:
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