  A computably enumerable equivalence relation (ceer) $X$ is called self-full
if whenever $f$ is a reduction of $X$ to $X$ then the range of $f$ intersects
all $X$-equivalence classes. It is known that the infinite self-full ceers
properly contain the dark ceers, i.e. the infinite ceers which do not admit an
infinite computably enumerable transversal. Unlike the collection of dark
ceers, which are closed under the operation of uniform join, we answer a
question from \cite{joinmeet} by showing that there are self-full ceers $X$ and
$Y$ so that their uniform join $X\oplus Y$ is non-self-full. We then define and
examine the hereditarily self-full ceers, which are the self-full ceers $X$ so
that for any self-full $Y$, $X\oplus Y$ is also self-full: we show that they
are closed under uniform join, and that every non-universal degree in
$\textrm{Ceers}_{/{\mathcal{I}}}$ have infinitely many incomparable
hereditarily self-full strong minimal covers. In particular, every
non-universal ceer is bounded by a hereditarily self-full ceer. Thus the
hereditarily self-full ceers form a properly intermediate class in between the
dark ceers and the infinite self-full ceers which is closed under $\oplus$.
