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Radical classes of cyclically ordered groups. (English) Zbl 0662.06004
The notions of a radical class and a K-radical class of linearly ordered groups (introduced by C. G. Chehata and R. Wiegandt [Math. Rév. Anal. Numér. Théor. Approx. 20, 143-157 (1978; Zbl 0409.06008)] and by J. Jakubík [Math. Slovaca 38, 33-44 (1988; Zbl 0646.06012)] are extended to the case of abelian cyclically ordered groups. Denote by $${\mathcal R}$$ the lattice of all radical classes of abelian linearly ordered groups; $${\mathcal R}_{{\mathcal K}}$$ be the lattice of all K-radical classes of abelian linearly ordered groups; $${\mathcal R}_{{\mathcal C}}$$ be the lattice of all radical classes of abelian cyclically ordered groups; $${\mathcal R}_{{\mathcal K}{\mathcal C}}$$ be the lattice of all K-radical classes of abelian cyclically ordered groups. In the paper under review order properties of $${\mathcal R}_{{\mathcal C}}$$ and $${\mathcal R}_{{\mathcal K}{\mathcal C}}$$ and relations between $${\mathcal R}$$, $${\mathcal R}_{{\mathcal K}}$$, $${\mathcal R}_{{\mathcal C}}$$ and $${\mathcal R}_{{\mathcal K}{\mathcal C}}$$ are investigated. From the results: $${\mathcal R}_{{\mathcal C}}$$ contains atoms and dual atoms; $${\mathcal R}_{{\mathcal K}}$$ is isomorphic to $${\mathcal R}_{{\mathcal K}{\mathcal C}}$$ but not a sublattice of $${\mathcal R}_{{\mathcal K}{\mathcal C}}$$; $${\mathcal R}$$ is a retract of $${\mathcal R}_{{\mathcal C}}$$.
Reviewer: M.Harminc

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
##### Keywords:
radical class; abelian cyclically ordered groups
Full Text:
##### References:
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