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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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