Twisted superspace on a lattice

We propose a new formulation which realizes exact twisted supersymmetry for all the supercharges on a lattice by twisted superspace formalism. We show explicit examples of N=2 twisted supersymmetry invariant BF and Wess-Zumino models in two dimensions. We introduce mild lattice non-commutativity to preserve Leibniz rule on the lattice. The formulation is based on the twisted superspace formalism for N=D=2 supersymmetry which was proposed recently. From the consistency condition of the non-commutativity of superspace, we find an unexpected three-dimensional lattice structure which may reduce into two-dimensional lattice where the superspace describes semilocally scattered fermions and bosons within a double size square lattice.