# superMatrixParity -- parity of a super Matrix.

## Synopsis

• Usage:
N = superMatrixParity(SM, R, L)
• Inputs:
• Outputs:
• N, , 0 for even, 1 for odd and-1 for Nonhomogeneous

## Description

This function works only when the entries of the even blocks are numbers, and those of odd blocks are formed by odd generators. Let $A^{p|q}=Ax_1\oplus \cdots \oplus Ax_p \oplus Ae_1\oplus\cdots\oplus Ae_q$ be a free module over $A$, where $x_i$s are even and $e_j$s are odd generators. A morphism $T:A^{p|q}\rightarrow A^{r|s}$ has a matrix representation. Denote the matrix by $T$ then we have $T=\begin{pmatrix} T1&T2\\ T3&T4\end{pmatrix}$.

The matrix (morphism) $T$ is said to be even (odd) if the blocks $T_1$ and $T_4$ are even, and $T_2$ and $T_3$ are odd ($T_1$ and $T_4$ are odd, and $T_2$ and $T_3$ are even). Note that if $A$ is an algebra, i.e., it doesn't have odd involution, then to have an even (odd) matrix $T$, we should have $T_3=0$ and $T_2=0$ ($T_1=0$ and $T_4=0$). Example R1 = QQ[x_0..x_3]; R2 = QQ[z_0..z_2]; R = superRing(R1, R2); D1 = matrix{{x_0, x_1}, {x_2, x_3}}; D2 = matrix{{z_0, z_1}, {x_0*z_0, x_1*z_1}}; D3 = matrix{{z_2*x_3, z_1}, {z_0, z_2*x_2}}; D4 = matrix{{x_1, x_3}, {x_0, x_2+x_3}}; SM = superMatrixGenerator(D1, D2, D3, D4); superMatrixParity(SM, R, {z_0, z_1, z_2})