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K-radical classes of Abelian linearly ordered groups. (English) Zbl 0646.06012
For a linearly ordered group G let c(G) denote the chain of all convex subgroups of G. A radical class X of abelian linearly ordered groups is said to be a K-radical class if whenever $$G_ 1\in X$$ and $$G_ 2$$ is an abelian linearly ordered group with $$c(G_ 2)\cong c(G_ 1)$$, then $$G_ 2\in X$$. It is shown that the set $$R_ K$$ of all K-radical classes X forms a complete sublattice of the lattice of all radical classes of abelian linearly ordered groups. For the least element in $$R_ K$$ of all K-radical classes X forms a complete sublattice of the lattice of all radical classes of abelian linearly ordered groups. For the least element in $$R_ K$$ containing X as a subclass a constructive description is given, as for the join operation in the lattice $$R_ K$$. Furthermore it is proved that in $$R_ K$$ no atom or dual atom exist. Finally, hereditary radical classes are studied: they are defined as closed with respect to isomorphisms and satisfying the condition that $$H\in X$$ whenever $$H\in c(G)$$ for some $$G\in X$$.
Reviewer: H.Mitsch

MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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