Tableaux problem 45

45.[Fom1994a]*sec. 2.3. 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 bilinear product on $V$ that turns it into a graded algebra with identity 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 us add that the product must be associative. If this product exists, it allows the graded linear operator $D$ to be used to construct a dual graded graph by the construction in the citation.