Various disconnectivities of spaces and projectabilities of ℓ-groups

Arch denotes the category of archimedean ℓ-groups and ℓ-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ℓ-group and demonstrate that the class of αcc-projectable ℓ-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W-the category of Arch-objects with designated weak unit and ℓ-homomorphisms that preserve the weak unit-and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ℓ-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps). © 2012 Springer Basel AG.