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Product radical classes of \(l\)-groups. (English) Zbl 0773.06019
Let \(\mathcal R\) be a radical class of \(\ell\)-groups; if \(\mathcal R\) is closed with respect to direct products then it is said to be a product radical class. In the present paper the properties of product radical mappings are studied. It is proved that the family of all product radical classes is a complete lattice. Under the usual definition of multiplication of radical classes, the following formulas are valid: \({\mathcal U}\cdot\bigl(\bigwedge_{\lambda\in\Lambda}{\mathcal I}_ \lambda\bigr)=\bigwedge_{\lambda\in\Lambda}({\mathcal U}\cdot{\mathcal I}_ \lambda)\), \({\mathcal U}\cdot\bigl(\bigvee^ n_{i=1}{\mathcal I}_ i\bigr)=\bigvee^ n_{i=1}({\mathcal U}\cdot{\mathcal I}_ i\bigr)\), where \({\mathcal U}\), \({\mathcal I}_ \lambda\) and \({\mathcal I}_ i\) are product radical classes. The polar closure operator and the operator of completion on the family of all product radical classes are dealt with. A homogeneity condition for \(\ell\)-groups, defined by means of the notion of product radical classes is studied.

MSC:
06F15 Ordered groups
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