does not affect the charges of the tensor networks, because the charge of $|\alpha
_{l})\otimes _{g}(\alpha _{l}|$ is zero. Therefore, the general parafermionic MPS is expressed as $$\begin{aligned}
|\psi \rangle  &=&\mathcal{C}(\mathbf{C}_{a}\otimes _{g}\mathbf{A}[1]\otimes
_{g}\mathbf{A}[2]\otimes _{g}\cdots \otimes _{g}\mathbf{A}[L])  \notag \\
&=&\sum_{i_{1}..i_{N}}\left( C_{a}^{T}A^{[i_{1}]}\cdots A^{[i_{L}]}\right)
|i_{1}\cdots i_{L}\rangle ,\end{aligned}$$where $\mathbf{C}_{a}=\sum_{\gamma \delta }C_{a,\gamma \delta }(\gamma
_{1}|\otimes _{g}|\delta _{L})$ is the closure tensor and different choices of $\mathbf{C}_{a}$ just result in the different charges of the closed MPS wave functions[@XuZhang]. It should be emphasized that, unlike the Majorana fermion chains, the periodic boundary condition can not reconcile with the algebras of parafermions, so the periodic boundary condition for parafermion chains does not exist. How to define the Hamiltonian for closed boundary conditions has been specifically discussed[XuZhang,Alexandradinata]{}.

Classification of $\mathbb{Z}_{p}$ parafermionic MPS with irreducibility
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Irreducibility is the most important property for a general MPS, because the irreducible MPS determines the major physical properties of the system. Irreducible forms of bosonic and fermionic MPS have been constructed, the concept of irreducibility of fermionic MPS is quite different from that of bosonic MPS. For the fermionic MPS, there are two types of irreducible fermionic MPS[@Bultinck2017; @Fidkowski2011], one is called even type with the local matrices with the block structures: 