We consider the anti-symmetrization of the half-shuffle on words, which we
call the 'area' operator, since it corresponds to taking the signed area of
elements of the iterated-integral signature. The tensor algebra is a so-called
Tortkara algebra under this operator. We show that the iterated application of
the area operator is sufficient to recover the iterated-integral signature of a
path. Just as the "information" the second level adds to the first one is known
to be equivalent to the area between components of the path, this means that
all the information added by subsequent levels is equivalent to iterated areas.
On the way to this main result, we characterize (homogeneous) generating sets
of the shuffle algebra. We finally discuss compatibility between the area
operator and discrete integration and stochastic integration and conclude with
some results on the linear span of the areas of areas.Comment: added examples and remarks, corrected semimartingale/martingale par