On the supersymmetric $N=1$ and $N=(1,1)$ spectral data : A multiplicativity property

We show that there are six different choice of tensor product of supersymmetric N=(1,1) spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We also show that the procedure of extending a supersymmetric N=1 spectral data to N=(1,1) spectral data respects only one tensor product among these. We refer this as the multiplicativity property of the extension procedure. Therefore, if we demand that the extension procedure is multiplicative then there is a unique choice of tensor product of N=(1,1) spectral data.