proofs as [@Ho2 Theorem 2.1] and [@Ho2 Theorem 3.1], $\calh_M$ is a bialgebra. By the definition of $\msh$ and $\Delta$, $\calh_M$ is connected filtered with the submodules $\calh_M^{(n)}, n\geq 0$. Then $\calh_M$ is automatically a Hopf algebra by  [@F-G Proposition 5.3], for example.

We prove the following property for later applications.

For $k\geq 1$, let $\Sigma_k$ be the permutation group on $\{1,\cdots,k\}$. For $\vec{a}=(a_1,\cdots,a_k)\in M^k \subseteq \calh_M$, define $\sigma(\vec{a})=(a_{\sigma(1)},\cdots,a_{\sigma(k)})$ and define $\vec{a}^{(\Sigma_k)}=\sum_{\sigma\in \Sigma_k} \sigma(\vec{a}).$ Then for $a_{k+1}\in M$, we have $$\vec{a}^{(\Sigma_k)} \msh (a_{k+1}) = (a_1,\cdots,a_k,a_{k+1})^{(\Sigma_{k+1})} + \sum_{i=1}^k (a_1,\cdots,a_ia_{k+1},\cdots,a_{k})^{(\Sigma_k)}
\mlabel{eq:permshuf}$$ where in the sum, $\sigma(a_1,\cdots,a_ia_{k+1},\cdots,a_k)=(a_{\sigma(1)},\cdots,a_{\sigma(i)}a_{k+1},\cdots,a_{\sigma(k)}).$

By the quasi-shuffle relation in Eq. (), we have $$\begin{aligned}
\lefteqn{\vec{a}^{(\Sigma_k)} \msh (a_{k+1})= \sum_{\sigma\in
\Sigma_k}
\Big((a_{\sigma(1)},\cdots,a_{\sigma(k)},a_{k+1})}\\
&& + \sum_{i=1}^k (a_{\sigma(1)},\cdots,a_{\sigma(i)},a_{k+1},a_{\sigma(i+1)},\cdots,a_{k})
 + \sum_{i=1}^k (a_{\sigma(1)},\cdots,a_{\sigma(i)}a_{k+1},\cdots,a_{\sigma(k)})\Big),\end{aligned}$$ hence the proposition. It also follows from the Partition Identity of Hoffman  whose proof only needs the quasi-shuffle relation , or the Bohnenblust-Spitzer formula for Rota-Baxter algebras .

Renormalized directional multiple zeta values
=============================================

We now introduce directional regularized MZVs and the corresponding Hopf algebra. We then show that the directional regularized MZVs have Laurent series expansion with log coefficients, giving an algebra homomorphism from the Hopf algebra to Laurent series. This allows us to apply the algebraic Birkhoff decomposition in Theorem  to 