if $|i|=2$, we can express the local matrices into more compact form: $$A^{[i]}=Y_{3}^{|i|}\otimes a^{[i]}.$$

To obtain the irreducible MPS, it should be guaranteed that $A^{[i]}$ must have no irreducible invariant subspace commuting with $Q_{3}$. If there was such an invariant subspace corresponding to the projector $\tilde{P}$, it should have the form $\tilde{P}=\text{diag}(\tilde{P}_{0},\tilde{P}_{1},\tilde{P}_{2})$. According to $A^{[i]}\tilde{P}=\tilde{P}A^{[i]}\tilde{P}$, it further satisfies $$a^{[i]}\tilde{P}_{0}=\tilde{P}_{0}a^{[i]}\tilde{P}_{0},a^{[i]}\tilde{P}_{1}=\tilde{P}_{1}a^{[i]}\tilde{P}_{1},a^{[i]}\tilde{P}_{2}=\tilde{P}_{2}a^{[i]}\tilde{P}_{2},$$for $\forall |i|=0$. To exclude such a situation, we must impose the necessary condition that the charge-0subalgebra spanned by all $\{a^{[i_{1}]}\cdots a^{[i_{p}]}\}$ with $\forall
p\in \mathbb{N}$ and $\sum_{l=1}^{p}|i_{l}|=0$ is simple. In the following, when we mention the charge-0subalgebra, it has the same definition, but it is the simple matrix algebra with different dimension.

So the local matrices $A^{[i]}$ are irreducible if $A^{[i]}$ can be gauge transformed into $Y_{3}^{|i|}\otimes a^{[i]}$ and the charge-0 sub-algebra is a simple matrix algebra. These conditions imply that $A^{[i]}$ span a non-trivial simple $\mathbb{Z}_{3}$ graded algebra. The graded center consists of multiples of $\mathbbm{1},Y_{3} $, and $Y_{3}^{2}$. Taking the trivial simple $\mathbb{Z}_{3}$ graded algebra of MPS into consideration, we obtain the conclusion that a $\mathbb{Z}_{3}$ parafermion MPS is irreducible iff $A^{[i]}$ span a simple $\mathbb{Z}_{3}$ graded algebra.

Topological order in $\mathbb{Z}_{3}$ parafermion MPS
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The characteristic properties of 