$$A^{[i]}=\left[ 
\begin{array}{cc}
a_{0}^{[i]} & 0 \\ 
0 & a_{1}^{[i]}\end{array}\right] ,|i|=0;\text{ }A^{[i]}=\left[ 
\begin{array}{cc}
0 & a_{0}^{[i]} \\ 
a_{1}^{[i]} & 0\end{array}\right] ,|i|=1,  \label{evenZ2}$$where the sub-block matrices can *not* be equal under gauge transformations. These irreducible matrices span the even type simple $\mathbb{Z}_{2}$ graded matrix algebra with the center consisting of multiples of the identity. So it is as simple as the ungraded algebra. While the other is called the odd type which can be gauge transformed into $$A^{[i]}=\left[ 
\begin{array}{cc}
a^{[i]} & 0 \\ 
0 & a^{[i]}\end{array}\right] ,|i|=0;\text{ }A^{[i]}=\left[ 
\begin{array}{cc}
0 & a^{[i]} \\ 
a^{[i]} & 0\end{array}\right] ,|i|=1.  \label{oddZ2}$$Then we can be further expressed into a more compact form: $$A^{[i]}=Y_{2}^{|i|}\otimes a^{[i]},\text{ }|i|=0,1,$$where $Y_{2}$ has been defined in Eq. (\[Y\]). Thus $A^{[i]}$ of irreducible MPS span an odd type simple $\mathbb{Z}_{2}$ graded algebra with the center consisting of multiples of $\mathbbm{1}$ and $Y_{2}$, characterizing the non-trivial topological phase with unpaired Majorana zero modes at the edges of one-dimensional systems. These two types of irreducible MPS represent the $\mathbb{Z}_{2}$ classification of one-dimensional interacting fermionic systems[@Fidkowski2011; @Turner2011].

Expanding the analysis of irreducibility, we can derive all irreducible $\mathbb{Z}_{p}$ parafermionic MPS, and the different algebras spanned by the local matrices correspond to distinct gapped phases. Therefore, we 