I'm pretty sure that use of the word "antiproduct" is unrelated to this use. Here, the "geometric antiproduct" is defined as the dual of the "geometric product"

I'm pretty sure that use of the word "antiproduct" is unrelated to this use. Here, the "geometric antiproduct" is defined as the dual of the "geometric product". I haven't been paying much attention, but I hope (1) that there's some kind of lattice structure here, and (2) that both the product and the antiproduct have strong geometric intuitions.