In the projective plane, there are 3 reference points
a1, a2, a3. We then get three reference
lines a1a2, a1a3,
a2a3. Finally, the plane itself is represented by the element a1a2a3. A general
point p in the plane is a linear combination of the reference points,
p=c1a1+c2a2+c3a3
where c1,c2,c3 are numerical weights.
Similarly, a general line is a linear combination of the reference lines.
There is only one planar (grade 3) reference element
a1a2a3 and so a general element of grade 3
is of the form,
W=c123a1a2a3
where c123 is a numerical weight. In other words, all the grade 3
elements are proportional to one another. In this sense, the elements
of grade 3 are like real numbers. The difference is that
ordinary numbers are scalars; they don't change under linear transformations,
whilst the new numbers get multiplied by the determinant of the linear
transformation. I call the elements of grade 3 pseudonumbers.
It turns out that the pseudonumbers are so like ordinary numbers that they
naturally occur as weights in the algebra. So, the coefficients ck
that were said to be numerical weights, can in fact be numbers or
pseudonumbers. Here is an illustration of how this can happen. Suppose
we form the product of two lines a1a2 and
a1a3. In Whitehead's algebra, the result of
multiplying the lines is to obtain the point at their intersection.
The algebra gives,
(a1a2)(a1a3)
=(a1a2a3)a1
which is the point a1 at the intersection of the pair of lines,
weighted by the pseudonumber a1a2a3. In
earlier computer implementations of the algebra, I did not see how to handle
the different types of numerical coefficient, so I arbitrarily set
a1a2a3=1 which is not correct, but doesn't
cause any trouble unless one starts to work with transformations which
have non-unity determinant; only then does the pseudo character of
pseudonumbers show up. (I'm not the only one who has had difficulty
handling the pseudonumbers, Whitehead himself also sets
a1a2a3=1 towards the end of article 98
of his book A Treatise on Universal Algebra,
and Barnabei, Brini and Rota restricted their paper on the
Exterior Calculus of Invariant Theory
to the group of transformations with unity determinant.) In the new
GAP implementation of Whitehead's algebra, the pseudonumber
a1a2a3=z where z is the indeterminate
of a univariate polynomial. This means that the coefficients ck
in Whitehead's algebra can be (Laurent) polynomials such as 3z^2-2z+4-8z^-1.
This is fine because the polynomials form a ring and mathematicians
have figured out that elements of a ring can be generalized coefficients
of a vector space (it's just that a vector space with ring coefficients
goes by the name of a module). The following program implements
the module over a polynomial ring version of Whitehead's algebra
and the diary file shows some examples of its use.
File Last Modified Description
Whiteheadv3.g 15-Oct-2006 Program to set up Whitehead's algebra in GAP diary.txt 15-Oct-2006 Log of GAP session to test the implementation of Whitehead's algebra