Injectivity of Archimedean \(\ell \)-groups with order unit
Let \(\mathbf W\) be the category of Archimedean \(\ell\)-groups with specified weak order unit and unit-preserving \(\ell\)-group homomorphisms. In unpublished notes (circa 1985), Hager proved by a lengthy argument that \(\mathbf W\) has only trivial injectives. The main contribution of the present paper consists in presenting two different proofs of the mentioned result, which are shorter and simpler than Hager's previous argument. Further, the authors deal with two subcategories of \(\mathbf W\), which are denoted as \(\mathbf{SW}\) and \(\mathbf W^*\). The subcategory \(\mathbf{SW}\) is given by the \(A\in \mathbf W\) for which the order unit is singular; the subcategory \(\mathbf W^*\) is given by the objects whose order unit is strong. It is proved that \(\mathbf{SW}\) has only trivial injectives and that \(\mathbf W^*\) has injective hulls.
scientific article; zbMATH DE number 5721490