  We examine the degree structure $\mathbf{ER}$ of equivalence relations on
$\omega$ under computable reducibility. We examine when pairs of degrees have a
join. In particular, we show that sufficiently incomparable pairs of degrees do
not have a join but that some incomparable degrees do, and we characterize the
degrees which have a join with every finite equivalence relation. We show that
the natural classes of finite, light, and dark degrees are definable in
$\mathbf{ER}$. We show that every equivalence relation has continuum many
self-full strong minimal covers, and that $\mathbf{d}\oplus \mathbf{Id_1}$
needn't be a strong minimal cover of a self-full degree $\mathbf{d}$. Finally,
we show that the theory of the degree structure $\mathbf{ER}$ as well as the
theories of the substructures of light degrees and of dark degrees are each
computably isomorphic with second order arithmetic.
