| 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:
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62 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
1073-1083 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2012-11-10T21:45:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143045 |
| . |
| Reference:
|
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