  We identify the free half shuffle algebra of Sch\"utzenberger (1958) with an
algebra of real-valued functionals on paths, where the half shuffle emulates
integration of a functional against another. We then provide two, to our
knowledge, new identities in arity 3 involving its commutator (area), and show
that these are sufficient to recover the Zinbiel and Tortkara identities of
Dzhumadil'daev (2007). We use these identities to prove that any element of the
free half shuffle algebra can be expressed as a polynomial over iterated areas.
Moreover, we consider minimal sets of iterated integrals defined through the
recursive application of the half shuffle on Hall trees. Leveraging the duality
between this set of Hall integrals and classical Hall bases of the free Lie
algebra, we prove using combinatorial arguments that any element of the free
half shuffle algebra can be written uniquely as a polynomial over Hall
integrals. We interpret this result as a structure theorem for streamed
information, loosely analogous to the unique prime factorisation of integers,
allowing to split any real valued function on streamed data into two parts: a
first that extracts and packages the streamed information into recursively
defined atomic objects (Hall integrals), and a second that evaluates a
polynomial function in these objects without further reference to the original
stream. The question of whether a similar result holds if Hall integrals are
replaced by Hall areas is left as an open conjecture. Finally, we construct a
canonical, but to our knowledge, new decomposition of the free half shuffle
algebra as shuffle power series in the greatest letter of the original alphabet
with coefficients in a sub-algebra freely generated by a new alphabet with an
infinite number of letters. We use this construction to provide a second proof
of our structure theorem.
