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)).