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