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Ton, Dao-Rong

The structure of a complete \(l\)-group. (English) Zbl 0819.06016

Czech. Math. J. 44, No. 2, 265-279 (1994).
The author describes the structure of complete \(\ell\)-groups in the following main theorem: Any complete \(\ell\)-group \(G\) is \(\ell\)- isomorphic to an ideal subdirect sum of integer groups and a complete \(\ell\)-group of \((\alpha, \aleph_ j)\) type.
Let us recall that an \(\ell\)-group \(G\) is an ideal subdirect sum of \(\ell\)-groups \(G_ \alpha\) if \(G\) is a subdirect sum of \(G_ \alpha\) and \(G\) is an ideal in their direct product. We denote the least cardinal \(\alpha\) such that \(| A|\leq \alpha\) for each bounded disjoint subset \(A\) of \(G\) by \(vG\), where \(| A|\) is the cardinal of \(A\). \(G\) is \(v\)-homogeneous if \(vH= vG\) for any convex \(\ell\)-subgroup \(H\neq \{0\}\) of \(G\). An \(\ell\)-group \(G\) is \(ic\)-homogeneous of \(\beta\) type if any nontrivial interval in \(G\) has the same cardinality \(\beta\). If \(\alpha\), \(\beta\) are two cardinal numbers, then \(G\) is said to be of type \((\alpha, \beta)\) if \(G\) is \(v\)-homogeneous of type \(\alpha\) (i.e., \(vG= \alpha\)) and \(ic\)-homogeneous of type \(\beta\).
Reviewer: B.F.Šmarda (Brno)

Cited in 2 Documents

MSC:

06F15 Ordered groups

Keywords:

\(v\)-homogeneous \(\ell\)-group; complete \(\ell\)-groups; ideal subdirect sum
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Full Text: DOI EuDML

References:

[1] M. Anderson, T. Feil: Lattice-Ordered Groups (An Introduction). D. Reidel Publishing Company, 1988. · Zbl 0636.06008
[2] S. J. Bernau: Unique representation of Archimedean lattice groups and normal Archimedean lattice rings. Proc. London Math. Soc. (3)15 (1965), 599-631. · Zbl 0134.10802 · doi:10.1112/plms/s3-15.1.599
[3] P. Conrad, D. Mcalister: The completion of a lattice ordered group. J. Austral. Math. Soc. 9 (1969), 182-208. · Zbl 0172.31601 · doi:10.1017/S1446788700005760
[4] P. Conrad: Lattice-Ordered Groups, Lecture Notes. Tulane University, 1970. · Zbl 0258.06011
[5] P. Conrad: The essential closure of an Archimedean lattice-ordered group. Duke Math. J. (1971), 151-160. · Zbl 0216.03104 · doi:10.1215/S0012-7094-71-03819-1
[6] P. Conrad: The hull of representable groups and \(f\)-rings. J. Austral. Math. Soc. 16 (1973), 385-415. · Zbl 0275.06025 · doi:10.1017/S1446788700015391
[7] L. Fuchs: Partially Ordered Algebraic Systems. Pergamon Press, 1963. · Zbl 0137.02001
[8] A. M. W. Glass, W. C. Hollad: Lattice-Ordered Groups (Advances and Techniques). Kluwer Academic Publishers, 1989. · Zbl 0705.06001
[9] K. Iwasawa: On the structure of conditionally complete lattice-groups. Japan J. Math. 18 (1943), 777-789. · Zbl 0061.03408
[10] J. Jakubík: Homogeneous lattice ordered groups. Czech. Math. J. 22(97) (1972), 325-337. · Zbl 0259.06016
[11] J. Jakubík: Cardinal properties of lattice ordered groups. Fundamenta Mathematicae 24 (1972), 85-98. · Zbl 0259.06015
[12] B. Z. Vulikh: Introduction to the Theory of Partially Ordered Space. Groningen, 1967. · Zbl 0186.44601
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