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 in the literature. It is known that every countable pca can be weakly embedded into $\mathcal{K}_2$, and we generalize this to arbitrary cardinalities by considering generalizations of $\mathcal{K}_2$ for larger cardinals. This emphasizes the central role of $\mathcal{K}_2$ in the study of pcas. We also show that $\mathcal{K}_2$ and its generalizations have strong completions.

翻译：我们研究了偏组合代数（pcas）的完备化问题，特别是克莱尼第二模型 $\mathcal{K}_2$ 及其推广形式的完备化。我们考察了文献中先前研究过的嵌入性与完备性的弱概念和强概念。已知每个可数 pca 均可弱嵌入到 $\mathcal{K}_2$ 中，我们通过考虑针对更大基数的 $\mathcal{K}_2$ 推广形式，将这一结果推广至任意基数情形。这突显了 $\mathcal{K}_2$ 在 pca 研究中的核心地位。我们还证明了 $\mathcal{K}_2$ 及其推广形式具有强完备化。