Yes, each product is related to its associated antiproduct by the GA equivalent of De Morgan’s laws with the small generalization that the abstract complement operator, the dual D in this case, isn’t necessarily an involution as it is in set theory and logic. \widehat P(x,y) := D^{-1}(P(D(x),D(y))) is exactly how I implement all antiproducts in Mathematica. (I refer to ∨ as the “exterior antiproduct”, or just the “antiwedge product”.) They are prominently featured at the top of my reference poster:
Something else you’ll see on this poster are the filled and empty bars in the table of basis elements. By recognizing that a basis element can simultaneously be characterized by its full dimensions (grade) and its empty dimensions (antigrade), the fundamental duality inherent in GA is made clear at the most basic level. Anything that can be called point-like by counting filled bars can also be called plane-like by counting empty bars, and vice-versa. I mentioned on Twitter yesterday that if you assign a 1 to filled bars and a 0 to empty bars, then each basis element in an n-D algebra has a unique n-bit code. In the nondegenerate case, disregarding sign, the geometric product then amounts to an XOR operation, and the geometric antiproduct amounts to an XNOR operation. You could also choose to assign a 0 to filled bars and a 1 to empty bars, and the two operations would then exchange places. The takeaway is that there is perfect symmetry, and there can be no statement made about points that is not also true, from an opposite perspective, about planes, and there can be no statement made about planes that is not also true, from an opposite perspective, about points. (And this naturally extends to elements of higher grade/antigrade in 5D+.)
Of course, the dual is a NOT operation on the basis elements, but we are free to choose a sign convention. I haven’t worked through it completely, but it would appear that we can actually choose which orientation is positive and which orientation is negative independently for each grade, which would mean that there are 32 valid dual operators in 4D GA, and only two of them are involutions. (It’s a much more manageable 8 if we require 1 \rightarrow \mathbf{I} and \mathbf{I} \rightarrow 1.) I prefer right and left complements (which are inverses of each other) because they are easy to understand and they have the nice relationships with reverse and antireverse that I highlighted in my blog post.
Thank you for your thoughtfulness in this regard. I agree that unity and commonality would have great benefits for everybody, but there are still people in your camp who are actively working against this goal. As recently as this week, De Keninck has advanced his personal vendetta to discredit my viewpoints, and that shit needs to stop immediately if you want cooperation to have any chance. While his GAME2020 lecture contained many interesting insights, it’s clear that his entire motivation was to disparage the vectors-are-points model by claiming there is geometric intuition that exists only in the vectors-are-planes model, and he continued to preach this in the video’s comments. If you understand duality as I have described it above, then you must also understand that any such one-sided advantage is impossible. I pointed out this general feature of GA while SIGGRAPH 2019 was still in session only to be subjected to ferocious backlash. Until I have reason to believe that De Keninck (and at least one other person that I won’t drag into this by name) is going to drastically change his attitude, I’m afraid I have no choice but to continue my GA research independently.
I expect you’ll hear some whining about how I’ve told some people that your SIGGRAPH course contains information that has since been proven to be untrue. A lot of people come to me with questions about GA, and if they happened to start learning with your course, they are usually struggling with what they now believe to be an unbreakable requirement that vectors must be planes. It takes a lot of effort for me to explain that both models are equally valid and they can keep vectors as points if they want to. But it’s especially difficult when members of the “antiproduct is crazy” tribe swoop in to trash everything I have to say whenever they get a Google alert for the words “geometric algebra”. Maybe you can understand if I preemptively inform people that new developments have proven that the SIGGRAPH course doesn’t paint the whole picture when a link is posted in my neighborhood so that, at very least, people can go into it knowing that some of the subject material has been superseded. I’m not telling anyone that the vectors-are-planes model is wrong, but only that the requirement that vectors be planes is not correct. Ideally, a notice could be attached to the course stating something along the lines of “In the time since this course was prepared, advances have been made in this subject, and it is now known that an equally valid model in which vectors are points can also be constructed.”