+ \big(a_1 b_1,   \vec{a}\,' \msh \vec{b}\,'\big)
\mlabel{eq:qshuf}$$ with the convention that $\vec{a}\,'=\bfone$ if $m=1$, $\vec{b}\,'=\bfone$ if $n=1$ and $(a_1b_1,\vec{a}\,'\msh \vec{b}\,')=(a_1b_1)$ if $m=n=1$.

Quasi-shuffle is also known as harmonic product  and coincides with the stuffle product  in the study of MZVs. Variations of the stuffle product have also appeared in  . See §  for further details. It is shown  to be the same as the mixable shuffle product  which is also called overlapping shuffles  and generalized shuffles , and can be interpreted in terms of Delannoy paths .

The following theorem is a simple generalization of [@Ho2 Theorem 2.1, 3.1] where $M$ has the extra condition of being a locally finite set to ensure the grading structure on $\calh_M$.

Let $M$ be a commutative semigroup. Equip $\calh_M$ with the submodules $\calh_M^{(n)}=\oplus_{i=0}^n \bfk\, M^i$, The quasi-shuffle product $\msh$, the deconcatenation coproduct $$\begin{aligned}
&&\Delta: \calh_M \to \calh_M \barot \calh_M,\\
 \Delta( a_1, \cdots ,  a_k)&=&1\barot (a_1,  \cdots ,  a_k)
+ \sum_{i=1}^{k-1} (a_1,  \cdots,  a_i)\barot (a_{i+1},  \cdots ,  a_k) \notag\\
&&+ (a_1,  \cdots ,  a_k) \barot 1
\mlabel{eq:coprod}\end{aligned}$$ and the projection counit $\vep: \calh_M \to \bfk$ onto the direct summand $\bfk \subseteq \calh_M$. Then $\calh_M$ is a commutative connected filtered Hopf algebra.

By the same 