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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.

References [Enhancements On Off] (What's this?)

  • [AC] M. Anderson and P. F. Conrad, Epicomplete $ \ell $-groups, Algebra Universalis 12 (1981), 224-241. MR 608666 (82d:06017)
  • [B] S. J. Bernau, Unique representation of archimedean lattice groups and normal archimedean lattice rings, Proc. London Math. Soc. (3) 16 (1966), 107-130. MR 0188113 (32:5554)
  • [BH I] R. N. Ball and A. W. Hager, Epimorphisms in archimedean $ \ell $-groups and vector lattices, Lattice-Ordered Groups, Advances and Techniques, (A. M. W. Glass and W. Charles Holland, Eds), Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1989.
  • [BH III] -, Epimorphisms in archimedean $ \ell $-groups and vector lattices with weak unit (and Baire functions), J. Austral. Math. Soc. (Ser. A) 48 (1990), 25-56. MR 1026835 (91g:06016)
  • [BH IV] -, Archimedean kernel distinguishing extensions of archimedean $ \ell $-groups with weak unit, Indiana J. Math. 29 (1987), 351-368. MR 971646 (89m:06022)
  • [BH V] -, Algebraic extensions and closure of archimedean $ \ell $-groups and vector lattices (in preparation).
  • [BKW] A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux reticules, Lecture Notes in Math., Vol. 608, Springer-Verlag, Berlin, Heidelberg and New York, 1977. MR 0552653 (58:27688)
  • [BL] R. D. Byrd and J. T. Lloyd, Closed subgroups and complete distributivity in lattice-ordered groups, Math. Z. 101 (1967), 123-130. MR 0218284 (36:1371)
  • [C1] P. F. Conrad, The essential closure of an archimedean lattice ordered group, Proc. London Math. Soc. 38 (1971), 151-160. MR 0277457 (43:3190)
  • [C2] -, The structure of an $ \ell $-group that is determined by its minimal prime subgroups, Ordered Groups, Lecture Notes in Pure and Appl. Math., Vol. 62, Dekker, New York, 1980.
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, reprinted as Graduate Texts in Math., 43, Springer-Verlag, Berlin, Heidelberg and New York, 1976. MR 0407579 (53:11352)
  • [HR] A. W. Hager and L. C. Robertson, Representing and ringi[ill]ng a Riesz space, Symposia Mathematica 21 (1977), 411-431. 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), 110-130. MR 0194880 (33:3086)
  • [HJ] M. Henriksen and D. G. Johnson, On the structure of a class of Archimedean lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. MR 0133698 (24:A3524)
  • [HVW] M. Henriksen, J. Vermeer, and R. G. Woods, Quasi-F covers of Tychonoff spaces, Trans. Amer. Math. Soc. 303 (1987), 779-803. MR 902798 (88m:54049)
  • [HS] H. Herrlich and G. Strecker, Category theory, Allyn & Bacon, Boston, Mass., 1973. MR 0349791 (50:2284)
  • [K] J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. (3) 13 (1963), 31-50. MR 0143837 (26:1387)
  • [L] R. Lagrange, Amalgamation and epimorphisms in $ m$-complete Boolean algebras, Algebra Universalis 4 (1974), 177-179. MR 0364045 (51:300)
  • [LZ] W. Luxemburg and A. Zaanen, Riesz spaces, Vol. I, North-Holland, Amsterdam, 1971. MR 0511676 (58:23483)
  • [MV] J. J. Madden and J. Vermeer, Epicomplete archimedean $ \ell $-groups via a localic Yosida theorem J. Pure Appl. Algebra (to appear).
  • [S] M. H. Stone, Boundedness properties in function lattices, Canad. J. Math. 1 (1949), 176-186. MR 0029091 (10:546a)

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