  A large family of relations among multiple zeta values may be described using
the combinatorics of shuffle and quasi-shuffle algebras. While the structure of
shuffle algebras have been well understood for some time now, quasi-shuffle
algebras were only formally studied relatively recently. In particular, Hoffman
gives a thorough discussion of the algebraic structure, including a choice of
algebra basis, and applies his results to produce families of relations among
multiple zeta values and their generalisations. In a recent preprint, Hirose
and Sato establish a family of relations coming from a new generalised shuffle
structure, lifting a set of graded relations established by the author to
genuine ungraded relations. In this paper, we define a commutative algebra
structure on the space of non-commutative polynomials in a countable alphabet,
generalising the shuffle-like structure of Hirose and Sato. We show that, over
the rational numbers, this generalised quasi-shuffle algebra is isomorphic to
the standard shuffle algebra, allowing us to reproduce most of Hoffman's
results on quasi-shuffle algebras. We then apply these results to the case of
multiple zeta values, reproducing several known families of results and
establishing several more.
