Atoning for past sins, not to mention compensating for the undeserved
obscurity of right contraction: to accompany
(X ^ Y) _| Z = X _| (Y _| Z) , Dorst (2.12)
I unveil before the expectant multitudes
X |_ (Y ^ Z) = (X |_ Y) |_ Z
--- proved immediately using Dorst (2.12) and
(X |_ Y)+ = Y+ _| X+ Dorst (2.8)
(unary "+" denotes reversion) --- left and right are actually isomorphic.
There seems to be a good deal more going on here than discussed in Dorst ---
Outer product is associative: X ^ (Y ^ Z) = (X ^ Y) ^ Z
(must be well-known, tho' Dorst doesn't mention it);
Contraction and inner "fat-dot" product are non-associative;
Inner "fat-dot" product does not conserve versors;
*** Contraction and outer product do conserve versors
(if both arguments are versor, so is the result).
The last conjectures look decidedly nontrivial to me, even for X ^ Y ;
as Lanco mentioned earlier, they're straightforward for blades (ie. both
versors and grators), but where does one go from there?
Finally, in my own computations with the Euclidean / projective algebra
Cl(n,0,1) I make heavy use of operator
X v Y == (X~ ^ Y~ )~
(unary "~" denotes dual, pace Dorst!), which is particularly important in
degenerate situations: on blades it corresponds to generic join, dually
to X ^ Y for generic meet. Dorst doesn't seem to recognise this; indeed,
I failed to make much sense of his final section on the topic at all.