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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 greatestelement. This requires a new kind of initial-segment result and a new extensionof embeddings result both in the hyperarithmetic setting.

The $\Sigma_2$ theory of $\mathscr{D}_h(\leq_h \mathcal{O})$ as an uppersemilattice with least and greatest element is decidable