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Radical subgroups of lattice ordered groups. (English) Zbl 0605.06013
Let G be an \(\ell\)-group and c(G) the lattice of all convex \(\ell\)- subgroups of G. A lattice ordered group \(H\in c(G)\) will be said to be a radical subgroup of G if, whenever \(G_ 1\in c(G)\) and \(H_ 1\in c(H)\) such that \(G_ 1\) is isomorphic to \(H_ 1\), then \(G_ 1\subseteq H\). The system R(G) of all radical subgroups of G is partially ordered by inclusion. If G has no nontrivial radical subgroups, then G is said to be r-homogeneous. G will be said to be totally r-inhomogeneous if, whenever \(0\neq H\in R(G)\), then there exists \(H_ 1\in R(G)\) such that \(\{0\}\subset H_ 1\subset H.\)
The main results of this paper concern the lattice R(G) for the case when G is a complete \(\ell\)-group. Let us mention the following existence results: For each \(\alpha >0\) there exists a proper class \(A_{\alpha}\) of mutually nonisomorphic complete \(\ell\)-groups such that for each \(G\in A_{\alpha}\), R(G) is isomorphic to the Boolean algebra \(2^{\alpha}\). For each ordinal \(\delta\) there is a complete \(\ell\)-group G such that R(G) is a chain isomorphic to \(\delta\). For each complete \(\ell\)-group G there exists a complete \(\ell\)-group \(G_ 1\) such that \(G\in R(G_ 1)\) and G is covered by \(G_ 1\) in the lattice \(R(G_ 1)\). There exists a proper class of mutually nonisomorphic totally r-inhomogeneous \(\ell\)- groups. The question whether there exists a complete totally r- inhomogeneous \(\ell\)-group \(G\neq \{0\}\) remains open. Some results on the lattice \({\mathcal R}_ c\) of all radical classes of complete \(\ell\)-groups are established.
Reviewer: F.Šik

MSC:
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
20E15 Chains and lattices of subgroups, subnormal subgroups
06B15 Representation theory of lattices
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References:
[1] P. Conrad: Lattice ordered groups. Tulane University 1970. · Zbl 0258.06011
[2] P. Conrad: K-radical classes of lattice ordered groups. Algebra. Proc. Conf. Carbondale (1980), Lecture Notes Math. 848, 1981, 186-207.
[3] P. Conrad: The lattice of all convex \(l\)-subgroups of a lattice ordered group. Czech. M. J. 15, 1965, 101-123. · Zbl 0135.06301
[4] L. Fuchs: Partially ordered algebraic systems. Oxford 1963. · Zbl 0137.02001
[5] C. Gofman: Remarks on lattice ordered groups and vector lattices I. Carathéodory functions. Trans. Amer. Math. Soc. 88, 1958, 107-120.
[6] J. Jakubík: Radical mappings and radical classes of lattice ordered groups. Symposia Math., 31 Academic Press, New York-London 1977, 451 - 477.
[7] J. Jakubík: Products of radical classes of lattice ordered groups. Acta Math. Univ. Comen. 39, 1980, 31-42.
[8] J. Jakubík: On K-radical classes of lattice ordered groups. Czech. Math. J. 33, 1983, 149 to 163.
[9] J. Jakubík: Projectable kernel of a lattice ordered group. Universal algebra and applications. Banach Center Publ. Vol. 9, 1982, 105-112.
[10] J. Jakubík: Cardinal properties of lattice ordered groups. Fund. Math. 74, 1972, 85 - 98. · Zbl 0259.06015
[11] R. S. Pierce: Complete Boolean algebras. Proc. Symp. Pure Math., Vol. 2, Amer. Math. Soc., 1961, 129-140. · Zbl 0101.27104
[12] F. Šik: Über die Beziehungen zwischen eigenen Sptizen und minimalen Komponenten einer 1-Gruppe. Acta math. Acad. Sci. Hung. 13, 1962, 83 - 93.
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