  We establish the decidability of the $\Sigma_2$ theory of
$\mathscr{D}_h(\leq_h \mathcal{O})$, the hyperarithmetic degrees below Kleene's
$\mathcal{O}$, in the language of uppersemilattices with least and greatest
element. This requires a new kind of initial-segment result and a new extension
of embeddings result both in the hyperarithmetic setting.
