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Convergences and complete distributivity of lattice ordered groups. (English) Zbl 0662.06005
Let G be an \(\ell\)-group and let \((G^ N)^+\) denote a positive cone of \(G^ N\). A convergence in G is a convex normal subsemigroup \(\alpha\) of \((G^ N)^+\) satisfying the following conditions: (i) if \(S\in \alpha\), then each subsequence of S belongs to \(\alpha\) ; (ii) if \(S\in (G^ N)^+\) and if each subsequence of S has a subsequence belonging to \(\alpha\), then S belongs to \(\alpha\) ; (iii) a constant sequence (g,g,g,...) with \(g\in G\) belongs to \(\alpha\) if and only if g is the neutral element of the group G.
Let Conv G be the set of all convergences in G partially ordered by set inclusion. The result of the paper says that if G is an Archimedean completely distributive \(\ell\)-group then Conv G possesses a greatest element.
Reviewer: M.Harminc

06F15 Ordered groups
Full Text: EuDML
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