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on Tableaux problem 45

The statement

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.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.