Singular Archimedean lattice-ordered groups

Abstract.

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if \( G = D(X, {\Bbb Z}) \), where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted \( {\beta}_s \), is obtained by contracting the maximum monoreflection \( \beta \) on W, and G is epicomplete in W _s precisely when G is laterally \( \sigma \)-complete; (iv) the maximum essential reflection on W _s , denoted \( \varepsilon _s \), is the contraction of the maximum essential reflection \( \varepsilon \) on W.