Given a graded vector space $V = \bigoplus_{i=0}^{\infty} V_i$ and a graded linear operator $D$ on it where $D(V_0) = \{0\}$ and $D(V_{i+1}) \subset D(V_{i-1})$, is it always possible to define a product on $V$ that turns it into a graded algebra for which $D$ is a derivation ($D(ab) = D(a)b + aD(b)$)?
It seems that one can always define such a product, but there is much freedom and no clear guarantee the product has any nice properties. So let me add that the product must be associative. How that constrains the construction is not at all clear to me. Is this a known result?