一句话：将 Kleene 第二模型 K2 的弱嵌入性质推广到任意基数，并证明其推广具有强完备化。
本文研究部分组合代数（pca）的完备化，重点关注 Kleene 第二模型 K2 及其推广。作者考察文献中已有的弱与强可嵌入性及完备化概念。已知每个可数 pca 均可弱嵌入 K2；本文通过构造适用于更大基数的 K2 推广，将该结果推广到任意基数，从而进一步凸显 K2 在 pca 研究中的核心地位。此外，论文证明 K2 及其推广均具有强完备化。工作属于可计算性理论与组合代数语义的交叉，采用数学证明与构造性方法，为 pca 的嵌入与完备化理论提供了统一视角。
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.