Initial segments of the degrees of ceers

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self full degrees. We show that the poset of finite strings of natural numbers, under the relation of being an initial segment, can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice finite strings of natural numbers, under the relation of being an initial segment, as an initial segment of the degrees of ceers with infinitely many classes