Some of the literature on the wedge product handles it all under a different set of names. In oriented projective geometry the wedge product is the 'join' opera

Some of the literature on the wedge product handles it all under a different set of names. In oriented projective geometry the wedge product is the 'join' operator. The antiwedge mentioned in this article is called the 'meet'. There are some other papers (by Gian Carlo Rota and others) which treat this operation in the exterior algebra directly, which I can find later. The meet operator is dual to the wedge product. Presumably the 'antigeometric product' in this article is along the same lines, but, like the geometric product, much harder to make sense of than the meet itself. I tried for a long time to find a good interpretation of the meet. It roughly corresponds to finding the intersection of two linear subspaces (the wedge product roughly corresponds to their linear span). I personally think the geometric product is the source of all the bafflement, and the real interesting object is the wedge product. I hope to write up a solid argument about this someday but haven't had the wherewithal. (I also suspect the geometric product is usually defined wrong! A product (xy) (x) should be y, not -y, the factors should cancel left to right in both terms. Afaict this simplifies a lot of formulas.)