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The Galois connection between weak torsion and sub-product classes of $$l$$-groups. (English) Zbl 0938.06013
An $$l$$-group $$G$$ is called a completely subdirect product of $$l$$-groups $$\{ G_{\alpha} \: \alpha \in I\}$$ if (i) $$G$$ is a subdirect product of these groups and (ii) $$\sum (G_{\alpha}; \alpha \in I)\subseteq G$$, where $$\sum$$ stands for the direct sum. Now, a class of $$l$$-groups is called a sub-product class if it is closed under taking convex $$l$$-groups and forming completely subdirect products.
On the other hand, a class of $$l$$-groups is a weak torsion class if it is closed under taking strong $$l$$-homotrophic images (i.e., the kernels are direct summands) and forming joins of convex $$l$$-subgroups.
The main result: There is a Galois connection between the weak torsion classes and the sub-product classes. This is a generalization of a similar result of J. Martinez [Trans. Am. Math. Soc. 259, 311-317 (1980; Zbl 0433.06016)] about torsion classes and torsion-free classes of $$l$$-groups.
##### MSC:
 06F15 Ordered groups
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##### References:
 [1] ANDERSON M.-FEIL T.: Lattice-Ordered Groups (An Introduction). D. Reidel Publishing Company, Boston, 1988. · Zbl 0636.06008 [2] CONRAD P.: Lattice-Ordered Groups. Tulane Lecture Notes, Tulane University, 1970. · Zbl 0258.06011 [3] GLASS A. M. W.-HOLLAND W. C: Lattice-Ordered Groups (Advances and Tech-niques). Kluwer Academic Publishes, London, 1989. [4] MARTINEZ J.: The fundamental theorem on torsion classes of lattice-ordered groups. Trans. Amer. Math. Soc. 259 (1980), 311-317. · Zbl 0433.06016 [5] TON, DAO-RONG.: Sub-product classes of lattice-ordered groups. Acta Math. Sinica 2(37) (1994), 224-229.
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