Epicomplete Archimedean groups and vector lattices
Authors:
Richard N. Ball and Anthony W. Hager
Journal:
Trans. Amer. Math. Soc. 322 (1990), 459478
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 latticeordered 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 realvalued 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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 A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux reticules, Lecture Notes in Math., Vol. 608, SpringerVerlag, Berlin, Heidelberg and New York, 1977. MR 0552653 (58:27688)
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 R. D. Byrd and J. T. Lloyd, Closed subgroups and complete distributivity in latticeordered groups, Math. Z. 101 (1967), 123130. MR 0218284 (36:1371)
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 P. F. Conrad, The essential closure of an archimedean lattice ordered group, Proc. London Math. Soc. 38 (1971), 151160. MR 0277457 (43:3190)
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 A. W. Hager and L. C. Robertson, Representing and ringi[ill]ng a Riesz space, Symposia Mathematica 21 (1977), 411431. MR 0482728 (58:2783)
 [HJe]
 M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110130. MR 0194880 (33:3086)
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 M. Henriksen and D. G. Johnson, On the structure of a class of Archimedean latticeordered algebras, Fund. Math. 50 (1961), 7394. MR 0133698 (24:A3524)
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 M. Henriksen, J. Vermeer, and R. G. Woods, QuasiF covers of Tychonoff spaces, Trans. Amer. Math. Soc. 303 (1987), 779803. MR 902798 (88m:54049)
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 H. Herrlich and G. Strecker, Category theory, Allyn & Bacon, Boston, Mass., 1973. MR 0349791 (50:2284)
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 J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3) 13 (1963), 3150. MR 0143837 (26:1387)
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 R. Lagrange, Amalgamation and epimorphisms in complete Boolean algebras, Algebra Universalis 4 (1974), 177179. MR 0364045 (51:300)
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 W. Luxemburg and A. Zaanen, Riesz spaces, Vol. I, NorthHolland, Amsterdam, 1971. MR 0511676 (58:23483)
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 J. J. Madden and J. Vermeer, Epicomplete archimedean groups via a localic Yosida theorem J. Pure Appl. Algebra (to appear).
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 M. H. Stone, Boundedness properties in function lattices, Canad. J. Math. 1 (1949), 176186. MR 0029091 (10:546a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009436036
PII:
S 00029947(1990)09436036
Keywords:
Archimedean group,
prime subgroup,
cozero set,
Boolean space,
Article copyright:
© Copyright 1990
American Mathematical Society
