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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

##### MSC:
 06F15 Ordered groups
##### Keywords:
lattice ordered group; convergence of sequences
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##### References:
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