α-projectable and laterally α-complete archimedean lattice-ordered groups with weak unit via topology

Let w be the category of archimedean lattice-ordered groups with weak order unit, comp the category of compact hausdorff spaces, and w y→ comp the yosida functor, which represents a w-object a as consisting of extended real-valued functions a ≤ d(y a) and uniquely for various features. this yields topological mirrors for various algebraic (w-theoretic) properties providing close analysis of the latter. we apply this to the sub-classes of α-projectable, and laterally α-complete objects, denoted p(α) and l(α), where α is a regular infinite cardinal or ∞. each w-object a has unique minimum essential extensions a ≤ p(α)a ≤ l(α)a in the classes p(α) and l(α), respectively, and the spaces y p(α)a and y l(α)a are recognizable (for the most part); then we write down what p(α)a and l(α)a are as functions on these spaces. the operators p(α) and l(α) are compared: we show that both preserve closure under all implicit functorial operations which are finitary. the cases of a = c(x) receive special attention. in particular, if (ω < α) l(α)c(x) = c(y l(α)c(x)), then x is finite. but (ω ≤ α) for infinite x, p(α)c(x) sometimes is, and sometimes is not, c(y p(α)c(x)).