i_{l}\cdots 0\rangle $ denotes the single-particle states. In order to encode the parafermion statistics into the Fock space, the graded tensor product $\otimes _{g}$ building many-body states is introduced as $$\begin{aligned}
\langle i_{1}i_{2}\cdots i_{L}| &=&\langle i_{L}|\otimes _{g}\cdots \otimes
_{g}\langle i_{2}|\otimes _{g}\langle i_{1}|,  \notag \\
|i_{1}i_{2}\cdots i_{L}\rangle &=&|i_{1}\rangle \otimes _{g}|i_{2}\rangle
\otimes _{g}\cdots \otimes _{g}|i_{L}\rangle ,\end{aligned}$$which describes the graded structure of Hilbert space mathematically. The crucial ingredient of the graded tensor product is the following isomorphism mapping $\mathcal{F}$: $$\begin{aligned}
\mathcal{F}(|i_{l}\rangle \otimes _{g}|j_{m}\rangle ) &\equiv &\omega
_{p}^{ij}|j_{m}\rangle \otimes _{g}|i_{l}\rangle ,  \notag \\
\mathcal{F}(\langle i_{l}|\otimes _{g}|j_{m}\rangle ) &\equiv &\bar{\omega}_{p}^{ij}|j_{m}\rangle \otimes _{g}\langle i_{l}|,\end{aligned}$$for $l<m$. The isomorphism $\mathcal{F}$ exchanges two nearby local Fock states, and the whole Fock space is a graded vector space, which is a generalization of super vector space of the fermions[@Bultinck2017]. Thus parafermion statistics is encoded into the Fock space by the isomorphism, which becomes crucial for the construction of the MPS wave functions.

Since a contraction is necessary for tensor networks, the homomorphism $\mathcal{C}$ has to be defined via a mapping $\mathbb{V}_{F}^{\ast }\otimes
_{g}\mathbb{V}_{F}\rightarrow \mathbb{C}$: $$\mathcal{C}\left( \langle i_{l}|\otimes _{g}|j_{l}\rangle \right) =\langle
i_{l}|j_{l}\rangle =\delta _{i_{l},j_{l}},$$which is nothing but the inner product and orthonormal. From the above propositions, the $p$-exclusion principle can be 