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Maximal convergences and minimal proper convergences in \(\ell\)-groups. (English) Zbl 0703.06011

From the authors’ introduction. “In this paper the notion of convergence on a lattice ordered group G will be applied in the same sense as in the first author’s earlier paper [ibid. 39(114), No.2, 232-238 (1989; Zbl 0681.06007)]. The system of all convergences on G will be denoted by Conv G. This system is partially ordered by inclusion. Assume that G is a direct product of lattice ordered groups \(G_ i\) (i\(\in I)\) with \(G_ i\neq \{0\}\) for each \(i\in I\). To each system \((\alpha_ i:\) \(i\in I)\), with \(\alpha_ i\in Conv G_ i\) for each \(i\in I\), there corresponds in a natural way an element \(\alpha\in Conv G\). Let S be the set of all \(\alpha\in Conv G\) which can be constructed in this way. Under the notation as above, \(\alpha\) is said to be the product of the system \((\alpha_ i:\) \(i\in I)\). The question arises whether a direct product of maximal elements \(\alpha_ i\) of Conv \(G_ i\) must be a maximal element of Conv G. Analogous questions were studied for topological groups and for convergence groups. In both these cases the answers are “No”. For the case of lattice ordered groups the following positive result will be established: (A) Let \(\alpha\in Conv G\), where \(G=\prod_{i\in I}G_ i\). Then the following conditions are equivalent: (i) \(\alpha\) is a maximal element of Conv G; (ii) there is a system \((\alpha_ i:\) \(i\in I)\) such that for each \(i\in I\), \(\alpha_ i\) is a maximal element of Conv \(G_ i\) and \(\alpha\) is the product of the system \((\alpha_ i:\) \(i\in I)\). The least element of Conv G will be denoted by d(G). If \(\alpha\in Conv G\) and \(\alpha\neq d(G)\) then \(\alpha\) will be said to be a proper convergence in G. A minimal proper convergence in G is called an atom of Conv G. The atoms of Conv G of an abelian lattice ordered group will be dealt with. The following results (B), (C) and (D) will be established: (B) Let \(\alpha\) be an atom of Conv G. Then the interval [d(G),\(\alpha\) ] of Conv G is a direct factor of Conv G. (C) Assume that Conv G has an atom. The following conditions are equivalent: (i) Conv G has a greatest element: (ii) each atom of Conv G has a pseudocomplement; (iii) there exists an atom in Conv G which has a pseudocomplement. (D) Let S be as above. Then each atom of Conv G belongs to S.”
Reviewer: F.Šik

MSC:

06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces

Citations:

Zbl 0681.06007
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