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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
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References:

[1] CONRAD P.: Lattice Ordered Groups. Tulane University, 1970. · Zbl 0258.06011
[2] EVERETT C. J., ULAM S.: On ordered groups. Trans. Amer. Math. Soc. 37, 1945, 208-216. · Zbl 0061.03406
[3] HARMINC M.: Sequential convergences on abelian lattice-ordered groups. Convergence Structures 1984. Mathematical Research, Band 24, Akademie-Verlag, Berlin; 1985, 153-158.
[4] HARMINC M.: The cardinality of the system of all sequential convergences on an abelian lattice ordered group. Czechoslov. Math. J. 37, 1987, 533-546. · Zbl 0645.06006
[5] HARMINC M.: Convergences on lattice ordered groups. Dissertation, Math. Inst. Slovak Acad. Sci., 1986. · Zbl 0581.06009
[6] JAKUBÍK J.: Distributivity in lattice ordered groups. Czech. Math. J. 22, 1977, 108-125.
[7] KONblTOB B. M.: Рєшєточно упорядочєнныє группы. Москва 1984.
[8] WEINBERG E. C.: Completely distributive lattice-ordered groups. Pacif. J. Math. 12, 1962, 1131-1148. · Zbl 0111.24301
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