  We study $Q$-polynomial distance-regular graphs from the point of view of
what we call descendents, that is to say, those vertex subsets with the
property that the width $w$ and dual width $w^*$ satisfy $w+w^*=d$, where $d$
is the diameter of the graph. We show among other results that a nontrivial
descendent with $w\ge 2$ is convex precisely when the graph has classical
parameters. The classification of descendents has been done for the 5 classical
families of graphs associated with short regular semilattices. We revisit and
characterize these families in terms of posets consisting of descendents, and
extend the classification to all of the 15 known infinite families with
classical parameters and with unbounded diameter.
