We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been studied before. By a result of Klop it is known that not every pca has a strong completion. The study of completions of $\mathcal{K}_2$ has as corollaries that weak and strong embeddings are different, and that every countable pca has a weak completion. We then consider generalizations of $\mathcal{K}_2$ for larger cardinals, and use these to show that it is consistent that every pca has a weak completion.

翻译：本文研究部分组合代数（pcas）的完备化，特别是克莱尼第二模型 $\mathcal{K}_2$ 及其推广形式的完备化。我们考察了先前研究中涉及的嵌入性与完备化的弱概念和强概念。根据克洛普的一个结果，已知并非所有部分组合代数都具有强完备化。对 $\mathcal{K}_2$ 完备化的研究可推得以下结论：弱嵌入与强嵌入是不同的，且每个可数部分组合代数都具有弱完备化。随后，我们考察 $\mathcal{K}_2$ 针对更大基数的推广形式，并利用这些结果证明“每个部分组合代数都具有弱完备化”这一命题是相容的。