Deformed N=1 supersymmetry

We consider a deformation of N=1 four dimensional Minkowski superspace where odd coordinates $\theta^{\alpha}$ do not anticommute. We define supersymmetric and associative star product and show how the remaining (anti)commutation relations among the superspace coordinates are modified. In particular, the even coordinates do not commute as well. We also study chiral and vector superfields and their interactions. Suprisingly we find that ordinary undeformed N=1 supersymmetric field theories are compatible with the deformed supersymmetry considered.