A note on initial segments of the enumeration degrees

We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: in fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: if a, b are enumeration degrees, with a total, and a < b, then in the degree interval (a, b), one can embed every countable partial order.