Take $S$ and $T$ two subsets of homogeneous elements of the associative noncommutative free algebra $F$ over a field $k$ generated by a set $X$, provided with a positive grading. 
To exclude pathologies $S$ and $T$ are contained in $\bar{F}.\bar{F}$, $\bar{F}$ denoting the augmentation ideal of $F$. 
If $S$ and $T$ are minimal generator sets for their respective generated ideals, is it the same true for the set $S.T$ of products of elements of $S$ and $T$?