obtain renormalized directional MZVs.

The Hopf algebra of directional regularized multiple zeta values
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We consider the commutative semigroup $$\frakM= \{{{\wvec{s}{r}}}\ \big|\ (s,r)\in \ZZ \times \RR_{>0}\}
\mlabel{eq:mbase}$$ with the multiplication $ {\wvec{s}{r}} {\wvec{s'}{r'}}={\wvec{s+s'}{r+r'}}.$ By Theorem , $$\calh_{\frakM}:=\sum_{k\geq 0} \CC\, \frakM^k,$$ with the quasi-shuffle product $\msh$ and the deconcatenation coproduct $\Delta$, is a connected filtered Hopf algebra. The same is true with the sub-semigroup $$\frakM^-=\{{\wvec{s}{r}}\ |\ (s,r)\in \ZZ_{\leq 0} \times \RR_{>0}\}.$$ For $w_i=\wvec{s_i}{r_i}\in \frakM,\ i=1,\cdots,k$, we use the notations $$\vec{w}=(w_1,\cdots,w_k)
=\wvec{s_1,\cdots,s_n}{r_1,\cdots,r_k}=\wvec{\vec{s}}{\vec{r}},\
{\rm where\ } \vec{s}=(s_1,\cdots,s_k), \vec{r}=(r_1,\cdots,r_k).$$ For $\vep\in \CC$ with ${\rm
Re}(\vep)<0$, define the [**directional regularized MZV**]{}: $$Z(\wvec{\vec{s}}{\vec{r}};\vep)=\sum_{n_1>\cdots>n_k>0}
\frac{e^{n_1\,r_1\vep} \cdots
    e^{n_k\,r_k\vep}}{n_1^{s_1}\cdots n_k^{s_k}}
\mlabel{eq:reggmzv}$$ It converges for any $\wvec{\vec{s}}{\vec{r}}$ and is regarded as the regularization of the [**formal MZV**]{} $$\zeta (\vec{s})= \sum_{n_1>\cdots>n_k>0} \frac{1}{n_1^{s_1} \cdots
    n_k^{s_k}}
\mlabel{eq:formgmzv}$$ which converges only when $s_i>0$ and $s_1>1$. It is related to the multiple polylogarithm $${\rm Li}_{s_1,\cdots,s_k}(z_1,\cdots,z_k)=\sum_{n_1>\cdots n_k>0}
    \frac{z_1^{n_1} \cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}}$$ by a change of variables $z_i=e^{r_i\vep}, 1\leq i\leq k$. As is well-known , the product of multiple polylogarithms as functions satisfies the quasi-shuffle (stuffle) relation of the nested sums. Therefore the product of regularized MZVs as functions also satisfies the quasi-shuffle relation: if $ \wvec{\vec{s}}{\vec{r}}\msh \wvec{\vec{s}\,'}{\vec{r}\,'}
=\sum \wvec{\vec{s}\,''}{\vec{r}\,''}$, then $$Z(\wvec{\vec{s}}{\vec{r}};\vep)Z(\wvec{\vec{s}\,'}{\vec{r}\,'};\vep)
= Z(\wvec{\vec{s}}{\vec{r}}\msh \wvec{\vec{s}\,'}{\vec{r}\,'};\vep)
:= \sum 