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Title: Ideal convergence and divergence of nets in $(\ell )$-groups (English)
Author: Boccuto, Antonio
Author: Dimitriou, Xenofon
Author: Papanastassiou, Nikolaos
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 62
Issue: 4
Year: 2012
Pages: 1073-1083
Summary lang: English
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Category: math
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Summary: In this paper we introduce the ${\mathcal I}$- and ${\mathcal I}^*$-convergence and divergence of nets in $(\ell )$-groups. We prove some theorems relating different types of convergence/divergence for nets in $(\ell )$-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 ${\mathcal I}^*$-convergence/divergence implies ${\mathcal I}$-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. (English)
Keyword: net
Keyword: $(\ell )$-group
Keyword: ideal
Keyword: ideal order
Keyword: $(D)$-convergence
Keyword: ideal divergence
MSC: 28B10
MSC: 28B15
MSC: 54A20
idZBL: Zbl 1274.28026
idMR: MR3010257
DOI: 10.1007/s10587-012-0064-z
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Date available: 2012-11-10T21:45:24Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/143045
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