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Some aspects of radical theory for fully ordered Abelian groups. (English) Zbl 0584.06010
The notions of radical class and semisimple class of linearly ordered groups were introduced by C. G. Chehata and R. Wiegandt [Math., Rev. Anal. Numér. Théor. Approximation, Math. 20, 143-157 (1978; Zbl 0409.06008)]. All linearly ordered groups dealt with in the present paper are assumed to be abelian. A class X of linearly ordered groups is said to be hereditary if, whenever \(G\in X\) and H is a convex subgroup of G, then \(H\in X\). It is proved that (i) each hereditary class determines a hereditary lower radical class, and (ii) each homomorphically closed class determines a homomorphically closed semisimple class. Further there are investigated radical classes and semisimple classes which are determined by conditions on skeletons of member groups. Several interesting examples are given.
Reviewer: J.Jakubík

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20E10 Quasivarieties and varieties of groups
20F60 Ordered groups (group-theoretic aspects)
06A06 Partial orders, general
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