This is how I’m thinking I may implement the naming convention in Grassmann in the future:
-
Right complement: metric independent Poincare dual so
I=x*complementright(x) -
Left complement: metric independent Poincare dual so
I=complementleft(x)*x - Hodge complement: metric dependent Hodge dual so det[g_x]I = x(\star x)
Note that x is a unit blade here. The reason for the last Hodge definition is very specific and is needed to satisfy \star\omega = \tilde\omega I, which is essential to be equivalent to complementright with the metric multiplied to it.
For usability, the Hodge dual will be ⋆ operator and the right complement will be ! for v0.3.2 Grassmann. This set of three definitions are the most compatible (for my purposes) with everything discussed so far.