can establish a complete classification of all one-dimensional parafermionic gapped phases, including the topological phases and conventional spontaneous symmetry breaking phases. The symmetry protected topological phases are not included, because no extra symmetry than the parafermionic charge is involved and $\mathbb{Z}_{p}$ symmetry is not sufficient to support symmetry protected topological phases. The general method is as follows. We first assume that all $A^{[i]}$ have an irreducible invariant subspace with the corresponding orthogonal projector $P_{0}$. Then we analyze all the commutation relations between the operators $P_{0}$ and $Q_{p}^{r}$, where $r $ is a positive integer. $P_{0}$ and $Q_{p}^{r}$ generate a space which contains all $\mathbb{Z}_{p}$ charge sectors. Finally, by using the invariant subspace projectors containing all charge sectors, we can determine the irreducible structures of $A^{[i]}$ and the corresponding algebras spanned by them. The different algebras associate with different gapped phases. In the following, several important cases are carried out in detail.

Irreducibility of $\mathbb{Z}_{3}$ parafermion MPS
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Now we first derive two types of irreducible MPS for $\mathbb{Z}_{3}$ parafermion chains[@XuZhang]. Without the lost of generality, we assume that there is an irreducible invariant subspace projector $P_{0}$ of local matrices $A^{[i]}$, i.e. $$A^{[i]}P_{0}=P_{0}A^{[i]}P_{0}.$$Because $A^{[i]}$ have a definite charge, $Q_{3}^{-1}A^{[i]}Q_{3}=\omega
^{|i|}A^{[i]}$, where 