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Ideal convergence and divergence of nets in (ℓ)-groups

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Abstract

In this paper we introduce the I- and I*-convergence and divergence of nets in (ℓ)-groups. We prove some theorems relating different types of convergence/divergence for nets in (ℓ)-group setting, in relation with ideals. We consider both order and (D)-convergence.

By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived.

We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, I-06123, Perugia, Italy

    Antonio Boccuto

  2. Department of Mathematics, University of Athens, Panepistimiopolis, Athens, 15784, Greece

    Xenofon Dimitriou & Nikolaos Papanastassiou

Authors
  1. Antonio Boccuto
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  2. Xenofon Dimitriou
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  3. Nikolaos Papanastassiou
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Correspondence to Antonio Boccuto.

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Supported by Universities of Perugia and Athens.

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Boccuto, A., Dimitriou, X. & Papanastassiou, N. Ideal convergence and divergence of nets in (ℓ)-groups. Czech Math J 62, 1073–1083 (2012). https://doi.org/10.1007/s10587-012-0064-z

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  • DOI: https://doi.org/10.1007/s10587-012-0064-z

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