Towards deformation quantization over a Z-graded base

The goal of this note is to describe a class of formal deformations of a symplectic manifold $M$ in the case when the base ring of the deformation problem involves parameters of non-positive degrees. The interesting feature of such deformations is that these are deformations “in $A_{\mathrm{\infty }}$-direction” and, in general, their description involves all cohomology classes of $M$ of degrees $\ge 2$.