Epicomplete Archimedean 𝑙-groups and vector lattices

Abstract

An object G G in a category is epicomplete provided that the only morphisms out of G G which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category A r c h {\mathbf {Arch}} , whose objects are the archimedean lattice-ordered groups (archimedean ℓ \ell -groups) and whose morphisms are the maps preserving both group and lattice structure ( ℓ \ell -homomorphisms). Recall that a space is basically disconnected if the closure of each cozero subset is open. Theorem. The following are equivalent for G ∈ A r c h G \in {\mathbf {Arch}} . (a) G G is A r c h {\mathbf {Arch}} epicomplete. (b) G G is an A r c h {\mathbf {Arch}} extremal suboject of D ( Y ) D(Y) for some basically disconnected compact Hausdorff space Y Y . Here D ( Y ) D(Y) denotes the continuous extended real-valued functions on Y Y which are finite on a dense subset. (c) G G is conditionally and laterally σ \sigma -complete (meaning each countable subset of positive elements of G G which is either bounded or pairwise disjoint has a supremum), and G G is divisible. The analysis of A r c h {\mathbf {Arch}} rests on an analysis of the closely related category W {\mathbf {W}} , whose objects are of the form ( G , u ) (G,u) , where G ∈ A r c h G \in {\mathbf {Arch}} and u u is a weak unit (meaning g ∧ u = 0 g \wedge u = 0 implies g = 0 g = 0 for all g ∈ G g \in G ), and whose morphisms are the ℓ \ell -homomorphism preserving the weak unit. Theorem. The following are equivalent for ( G , u ) ∈ W (G,u) \in {\mathbf {W}} . (a) ( G , u ) (G,u) is W {\mathbf {W}} epicomplete. (b) ( G , u ) (G,u) is W {\mathbf {W}} isomorphic to ( D ( Y ) , 1 ) (D(Y),1) . (c) ( G , u ) (G,u) is conditionally and laterally σ \sigma -complete, and G G is divisible.