Some unusual epicomplete Archimedean lattice-ordered groups

Hager, Anthony

doi:10.1090/S0002-9939-2015-12448-3

Some unusual epicomplete Archimedean lattice-ordered groups
HTML articles powered by AMS MathViewer

by Anthony W. Hager PDF

Proc. Amer. Math. Soc. 143 (2015), 1969-1980 Request permission

Abstract:

An Archimedean $l$-group is epicomplete if it is divisible and $\sigma$-complete, both laterally and conditionally. Under various circumstances it has been shown that epicompleteness implies the existence of a compatible reduced $f$-ring multiplication; the question has arisen whether or not this is always true. We show that a set-theoretic condition weaker than the continuum hypothesis implies “not”, and conjecture the converse. The examples also fail decent representation and existence of some other compatible operations.

References

Richard N. Ball, Truncated abelian lattice-ordered groups I: The pointed (Yosida) representation, Topology Appl. 162 (2014), 43–65. MR 3144659, DOI 10.1016/j.topol.2013.11.007
S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599–631. MR 182661, DOI 10.1112/plms/s3-15.1.599
Richard N. Ball and Anthony W. Hager, Epicomplete Archimedean $l$-groups and vector lattices, Trans. Amer. Math. Soc. 322 (1990), no. 2, 459–478. MR 943603, DOI 10.1090/S0002-9947-1990-0943603-6
Richard N. Ball and Anthony W. Hager, Epicompletion of Archimedean $l$-groups and vector lattices with weak unit, J. Austral. Math. Soc. Ser. A 48 (1990), no. 1, 25–56. MR 1026835, DOI 10.1017/S1446788700035175
Karim Boulabiar and Anthony Hager, $\ell$-group homomorphisms between reduced Archimedean $f$-rings, Algebra Universalis 62 (2009), no. 4, 329–337. MR 2670167, DOI 10.1007/s00012-010-0031-1
R. N. Ball, A. W. Hager, D. G. Johnson, and A. Kizanis, On the epicomplete monoreflection of an Archimedean lattice-ordered group, Algebra Universalis 54 (2005), no. 4, 417–434. MR 2218854, DOI 10.1007/s00012-005-1952-y
R. N. Ball, A. W. Hager, D. G. Johnson, and A. Kizanis, A theorem and a question about epicomplete Archimedean lattice-ordered groups, Algebra Universalis 62 (2009), no. 2-3, 165–184. MR 2661373, DOI 10.1007/s00012-010-0049-4
Paul Conrad and Jorge Martinez, Settling a number of questions about hyper-Archimedean lattice-ordered groups, Proc. Amer. Math. Soc. 109 (1990), no. 2, 291–296. MR 998733, DOI 10.1090/S0002-9939-1990-0998733-5
Paul Conrad, The essential closure of an Archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. MR 277457
Paul Conrad, The additive group of an $f$-ring, Canadian J. Math. 26 (1974), 1157–1168. MR 354487, DOI 10.4153/CJM-1974-108-3
Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961/62), 73–94. MR 133698, DOI 10.4064/fm-50-1-73-94
A. W. Hager and D. G. Johnson, Some comments and examples on generation of (hyper-)archimedean $\ell$-groups and $f$-rings, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. Fascicule Spécial, 75–100 (English, with English and French summaries). MR 2675722, DOI 10.5802/afst.1276
W Charles Holland. The Black Swamp Problem Book. 2003.
Anthony W. Hager and Lewis C. Robertson, Representing and ringifying a Riesz space, Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London, 1977, pp. 411–431. MR 0482728
Horst Herrlich and George E. Strecker, Category theory, 3rd ed., Sigma Series in Pure Mathematics, vol. 1, Heldermann Verlag, Lemgo, 2007. An introduction. MR 2377903
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
D. G. Johnson, On a representation theory for a class of Archimedean lattice-ordered rings, Proc. London Math. Soc. (3) 12 (1962), 207–225. MR 141685, DOI 10.1112/plms/s3-12.1.207
D. G. Johnson, A representation theorem revisited, Algebra Universalis 56 (2007), no. 3-4, 303–314. MR 2318213, DOI 10.1007/s00012-007-2000-x