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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.
06F15 Ordered groups
Full Text: EuDML
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[2] CONRAD P.: Lattice-Ordered Groups. Tulane Lecture Notes, Tulane University, 1970. · Zbl 0258.06011
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