On radical classes of Abelian linearly ordered groups.

*(English)*Zbl 0579.20034The author modifies the definition of the radical class of linearly ordered groups given in the radical theory developed by C. G. Chehata and R. Wiegandt in the sense of Kurosh-Amitsur [Math., Rev. Anal. Numér. Théor. Approximation, Math. 20, 143-157 (1978; Zbl 0409.06008)]. According to this modification, a nonempty subclass C of \({\mathcal G}_{\alpha}\), the class of all abelian linearly ordered groups, is a radical class of (i) C is closed with respect to homomorphisms, and (ii) if \(G\in {\mathcal G}_{\alpha}\) is a transfinite extension of linearly ordered groups belonging to C, then G belongs to C. The author then investigates the lattice \({\mathcal R}_{\alpha}\) of all radical classes in \({\mathcal G}_{\alpha}\) and shows that the results obtained differ essentially from the corresponding results concerning the lattice \({\mathcal T}\) of torsion classes of lattice ordered groups obtained by J. Martinez [Czech. Math. J. 25(100), 284-299 (1975; Zbl 0321.06020) and ibid. 26(101), 93-100 (1976; Zbl 0331.06009)] and also differ from the results about the lattice \({\mathcal R}\) of radical classes of lattice ordered groups obtained by the author [Symp. Math. 21, 451-477 (1977; Zbl 0368.06013)].

Reviewer: C.G.Chehata

##### MSC:

20F60 | Ordered groups (group-theoretic aspects) |

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

20E10 | Quasivarieties and varieties of groups |

08B15 | Lattices of varieties |

06B15 | Representation theory of lattices |

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##### References:

[1] | CHEHATA C. G., WIEGANDT R.: Radical theory for fully ordered groups. Mathematica (Cluj) 20 (43), 1979, 143-157. · Zbl 0409.06008 |

[2] | FUCHS L.: Partially Ordered Algebraic Systems. Pergamon Press, 1963. · Zbl 0137.02001 |

[3] | CONRAD P.: Lattice Ordered Croups. Tulane University 1970. |

[4] | JAKUBÍK J.: Radical mappings and radical classes of lattice ordered groups. Symposia mathem. 31, 1977, 451-477. · Zbl 0368.06013 |

[5] | MARTINEZ J.: Torsion theory for lattice ordered groups, I. Czech. Math. J. 25, 1975, 284-299. · Zbl 0321.06020 |

[6] | MARTINEZ J.: Torsion theory for lattice ordered groups, II. Czech. Math. J. 26, 1976, 93-100. · Zbl 0331.06009 |

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