$\{0\}\subseteq {F}_{0}A\subseteq \cdots \subseteq {F}_{i}A\subseteq \cdots \subseteq A,$

and suppose that

${\mathrm{g}\mathrm{r}}_{\bullet}^{F}A:=\underset{i\in {\mathbb{N}}_{0}}{\u2a01}{\mathrm{g}\mathrm{r}}_{i}^{F}A$

is the associated graded algebra of $A$, where ${\mathrm{g}\mathrm{r}}_{i}^{F}A:={F}_{i}A/{F}_{i-1}A$ and ${\mathrm{g}\mathrm{r}}_{0}^{F}A={F}_{0}A$.

If ${\mathrm{g}\mathrm{r}}_{\bullet}^{F}A$ is commutative, does it follow that ${F}_{i+j}A\subseteq {F}_{i}A\cdot {F}_{j}A$ for all $i,j\in {\mathbb{N}}_{0}$?