Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category $$\mathbf {APoSgr}_{\le }$$ APoSgr ≤ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$\mathscr {E}_{\le }$$ E ≤ -injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$\mathscr {E}_{\le }$$ E ≤ denote the class of morphisms $$h:A\longrightarrow B$$ h : A ⟶ B that preserve the compact elements and satisfy that $$h(a_1)\cdots h(a_n)\le h(b)$$ h ( a 1 ) ⋯ h ( a n ) ≤ h ( b ) always implies $$a_1\cdots a_n\le b$$ a 1 ⋯ a n ≤ b .

中文翻译：

---

偏序半群的前相干量子补全

正如完全格可以被视为偏序集的完成一样，量子也可以被视为偏序半群的完成。受对偏序集的著名 Frink 补全的研究的启发，很自然地考虑偏序半群情况下的“Frink”补全。为此，我们首先引入了偏序半群的前相干量子完备的概念，构造了偏序半群的三种类型的前相干量子完备的具体形式。此外，我们获得了偏序半群上的 Frink 完备是一个前相干量子完备的充分必要条件。最后，