Epicomplete Archimedean $l$-groups and vector lattices

Abstract:

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