| Abstract: | 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 $$mathbf {APoSgr}_{le }$$ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$mathscr {E}_{le }$$-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$mathscr {E}_{le }$$ denote the class of morphisms $$h:Alongrightarrow B$$ that preserve the compact elements and satisfy that $$h(a_1)cdots h(a_n)le h(b)$$ always implies $$a_1cdots a_nle b$$. |
