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


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.