# zbMATH — the first resource for mathematics

Locally conditioned radical classes of lattice-ordered groups. (English) Zbl 0767.06016
Let $$G$$ be an $$\ell$$-group, $$G(x)$$ be a convex $$\ell$$-subgroup of $$G$$ generated by $$x$$ and $$N_ x$$ be the intersection of all values of $$x$$ in $$G(x)$$. Let $$X$$ be an arbitrary class of archimedean $$\ell$$-groups and $$\text{Loc}(X)$$ be the class of all $$\ell$$-groups $$G$$ for which every local residue $$G(x)/N_ x$$ belongs to $$X$$.
In this paper, properties of $$\text{Loc}(X)$$ are investigated. It is proved that for a lot of classes $$X$$ of archimedean $$\ell$$-groups $$\text{Loc}(X)$$ is a torsion class.

##### MSC:
 06F15 Ordered groups 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
##### Keywords:
radical class; archimedean $$\ell$$-groups; torsion class
Full Text:
##### References:
 [1] M. Anderson, T. Feil: Lattice-Ordered Groups. Kluwer Publ., Dordrecht, Boston, 1988. · Zbl 0636.06008 [2] M. Anderson: Subprojectable and localy flat lattice-ordered groups. University of Kansas Dissertation, 1977. [3] R. Ball, P. Conrad, M. Darnel: Above and below subgroups of a lattice-ordered group. Trans. Amer. Math. Soc. 297 (1986), 1-40. · Zbl 0628.06013 [4] A. Bigard, P. Conrad, S. Wolfenstein: Compactly generated lattice-ordered groups. Math. Zeitschr. 107 (1968), 201-211. · Zbl 0182.36202 [5] A. Bigard, K. Keimel, S. Wolfenstein: Groupes et Anneaux Réticulés. Lecture Notes in Math. 608, Springer-Verlag, Berlin, Heidelberg, New York, 1977. [6] J. P. Bixler, M. Darnel: Special-valued $$\ell$$-groups. Alg. Univ. 22 (1986), 172-191. · Zbl 0582.06017 [7] P. Conrad: The lattice of all convex $$\ell$$-subgroups of a lattice-ordered group. Czech. Math. Jour. 15 (1965), 101-123. · Zbl 0135.06301 [8] P. Conrad: A characterization of lattice-ordered groups by their convex $$\ell$$-subgroups. Jour. Austral. Math. Soc. 7 (1967), 145-159, MR 35:5371. · Zbl 0154.27001 [9] P. Conrad: Epi-archimedean groups. Czech. Math. Jour. 24 (1974), 1-27. · Zbl 0319.06009 [10] P. Conrad: $$K$$-radical classes of lattice-ordered groups. Lecture Notes in Math. 848, Springer-Verlag, 1980, pp. 186-206. [11] P. Conrad, J. Martinez: Locally finite conditions on lattice-ordered groups. Czech. Math. Jour. 39 (1989), 432-444. · Zbl 0688.06011 [12] P. Conrad, J. Martinez: Signatures and $$S$$-discrete lattice-ordered groups. Alg. Univ · Zbl 0767.06015 [13] W. C. Holland: Varieties of lattice-ordered groups are torsion classes. Czech. Math. Jour. 29 (1979), 11-12. · Zbl 0432.06011 [14] G. O. Kenny: Lattice-Ordered Groups. University of Kansas Dissertation, 1975. [15] J. Martinez: The hyper-archimedean kernel sequence of a lattice-ordered group. Bull. Austral. Math. Soc. 10 (1974), 337-350. · Zbl 0275.06026 [16] J. Martinez: Torsion theory for lattice-ordered groups. Czech. Math. Jour. 25 (1975), 284-299. · Zbl 0321.06020 [17] J. Martinez: Pairwise-splitting lattice-ordered groups. Czech. Math. Jour. 27 (1977), 545-551. · Zbl 0378.06009 [18] J. Martinez: The closed subgroup problem for lattice-ordered groups. Archiv. der Math. 54 (1990), 212-224. · Zbl 0682.06011 [19] F. Šik: Zur theorie der halbgeordnete Gruppen. Czech. Math. Jour. 10 (1960), 400-424.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.