  Very little is known about the structure of the intersection of two
$k$-generated monoids of words, even for $k=3$. Here we investigate the case of
$k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be
non-trivially decomposed into at most $k$ words. We characterize the
intersection in the case of two $3$-maximal monoids.
