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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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