Definition of associated filtered algebra

Question

Consider the following fragment from wikipedia:

Can someone justify the choice of multiplication on this algebra? For me, it seems more natural define $$(x+ F_{n-1)})(y+ F_{m-1}):= xy + F_{n+m-2}$$

Why do they use $F_{n+m-1}$ instead of $F_{n+m+2}$?

Answer 1

Because the product of an element of $G_n$ and an element of $G_m$ must be in $G_{n+m}$, that is something of the form $z+F_{n+m-1}$ for $z\in F_{n+m}$.

Moreover, taking $xy+F_{n+m-2}$ would not even be well-defined: adding an element $u\in F_{n-1}$ to $x$ adds $uy\in F_{n+m-1}$ to the product, so $xy$ is really well defined up to something in $F_{n+m-1}$, not $F_{n+m-2}$.