Epicomplete Archimedean -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 in a category is *epicomplete* provided that the only morphisms out of which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category , whose objects are the archimedean lattice-ordered groups (archimedean -groups) and whose morphisms are the maps preserving both group and lattice structure (-homomorphisms). Recall that a space is *basically disconnected* if the closure of each cozero subset is open.

**Theorem.** *The following are equivalent for* .

(a) *is* *epicomplete*.

(b) *is an* *extremal suboject of* *for some basically disconnected compact Hausdorff space* . *Here* *denotes the continuous extended real-valued functions on* *which are finite on a dense subset*.

(c) *is conditionally and laterally* -*complete* (*meaning each countable subset of positive elements of* *which is either bounded or pairwise disjoint has a supremum*), *and* *is divisible*.

The analysis of rests on an analysis of the closely related category , whose objects are of the form , where and is a *weak unit* (meaning implies for all ), and whose morphisms are the -homomorphism preserving the weak unit.

**Theorem.** *The following are equivalent for* .

(a) *is* *epicomplete*.

(b) *is* *isomorphic to* .

(c) *is conditionally and laterally* -*complete, and* *is divisible*.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1990-0943603-6

Keywords:
Archimedean -group,
prime subgroup,
cozero set,
Boolean space,

Article copyright:
© Copyright 1990
American Mathematical Society