On the algebra of quasi-shuffles. (Q884990)

A commutative tri-dendriform algebra (CTD-algebra) is determined by two operations, denoted by \(\prec\) (left) and \(\cdot\) (dot), satisfying certain axioms. The aim of the paper is to show that the tensor module of the (non-constant) polynomial algebra with the quasi-shuffle algebra structure (called the stuffle algebra) is a free unital CTD-algebra. A non-commutative version of this is also proved in the paper. A quasi-shuffle algebra is in fact a CTD-bialgebra. The author proves a structure theorem for connected CTD-bialgebras.