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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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References:
[1] CHEHATA C. G., WIEGANDT R.: Radical theory for fully ordered groups. Mathematica - Rév. d’Anal. Numér. Théor. Approx. 20 (43), 1979, 143-157. · Zbl 0409.06008
[2] CONRAD P.: K-radical classes of lattice ordered groups. Conf. Algebra Carbondale, 1980, Lect. Notes in Mathematics 848, Springer Verlag 1981, 186-207.
[3] DARNEL M.: Closure operators on radicals of lattice ordered groups. Czech. Math. J. 37, 1987, 51-64. · Zbl 0624.06022
[4] FUKS L.: Častično uporjadočennye algebraičeskie sistemy. Moskva, 1972.
[5] GARDNER B. J.: Some aspects of radical theory for fully ordered abelian groups. The University of Tasmania, Technical Report No 203, 1985. · Zbl 0584.06010
[6] JAKUBÍK J.: On the lattice of radical classes of linearly ordered groups. Studia scient. mathem. Hungar. 19, 1981, 76-86. · Zbl 0465.06012
[7] JAKUBÍK J.: On the lattice of semisimple classes of linearly ordered groups. Čas. pěst. matem. 107, 1982, 183-190.
[8] JAKUBÍK J.: On K-radical classes of lattice ordered groups Czechoslov. Math J. 1983, 149-163.
[9] JAKUBÍK J.: On radical classes of abelian lattice ordered groups. Math. Slovaca 35, 1985, 141-154. · Zbl 0579.20034
[10] JAKUBÍKOVÁ M.: Hereditary radical classes of linearly ordered groups. Čas. pěst, matem. 107, 1982, 199-207.
[11] PRINGEROVÁ G.: Covering condition in the lattice of radical classes of linearly ordered groups. Math. Slovaca 33, 1983, 363-369. · Zbl 0519.06015
[12] PRINGEROVÁ G.: On semisimple classes of abelian linearly ordered groups. Čas. pěst. matem. 108, 1983, 40-52. · Zbl 0516.06013
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