y_{s,j}\Big(\tau_j\big(\tau_{-j}(\om_1)*y_{t,k}\om_2\big)\Big)
 + y_{t,k}\Big(\tau_k\big( y_{s,j}\om_1*\tau_{-k}(\om_2)\big)\Big) \\
 +y_{s+t,j+k}\Big(\tau_{j+k}\big(\tau_{-j}(\om_1)*\tau_{-k}(\om_2)\big)\Big).\end{gathered}$$ We call this multiplication the *stuffle product*.

Our $\fA$, $\fA^0$ and $\fA^1$ are related to $\QX$, $\QX_\cv$ and $\QY$ of [@Rac], respectively. See section \[sec:Rac\].

If we denote by $\fA_*^1$ the algebra $(\fA^1,*)$ then it is not hard to show that (cf. [@H2 Thm. 2.1])

The polynomial algebra $\fA_*^1$ is a commutative graded $\Q$-algebra.

Now we can define the subalgebra $\fA_*^0$ similar to $\fA_\sha^0$ by replacing the shuffle product by stuffle product. Then by induction on the lengths and using the series definition we can quickly check that for any $\om_1,\om_2\in \fA_*^0$ $$Z(\om_1)Z(\om_2)=Z(\om_1\ast \om_2).$$ This implies that

\[\*homo\] The map $Z:  \fA_*^0  \lra \C$ is an algebra homomorphism.

For $\om_1,\om_2\in \fA^0$ we will say that $$Z(\om_1\sha \om_2-\om_1*\om_2)=0$$ is a finite double shuffle (FDS) relation. It is known that even in level one these relations are not enough to provide all the relations among MZVs. However, it is believed that one can remedy this by considering RDS produced by the following mechanism. This was explained in detail in [@IKZ] when Ihara, Kaneko and Zagier considered MZVs where they call these extended double shuffle relations.

Combining Propositions \[\*homo\] and \[shahomo\] we can prove easily (cf. [@IKZ 