Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Epicomplete Archimedean $ l$-groups and vector lattices

Authors: Richard N. Ball and Anthony W. Hager
Journal: Trans. Amer. Math. Soc. 322 (1990), 459-478
MSC: Primary 46A40; Secondary 06F20, 46M15
MathSciNet review: 943603
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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.

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Keywords: Archimedean $ \ell $-group, prime subgroup, cozero set, Boolean space, $ C(Y)$
Article copyright: © Copyright 1990 American Mathematical Society