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

Antonio Boccuto; Xenofon Dimitriou; Nikolaos Papanastassiou

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 4, page 1073-1083
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Boccuto, Antonio, Dimitriou, Xenofon, and Papanastassiou, Nikolaos. "Ideal convergence and divergence of nets in $(\ell )$-groups." Czechoslovak Mathematical Journal 62.4 (2012): 1073-1083. <http://eudml.org/doc/246170>.

@article{Boccuto2012,
abstract = {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.},
author = {Boccuto, Antonio, Dimitriou, Xenofon, Papanastassiou, Nikolaos},
journal = {Czechoslovak Mathematical Journal},
keywords = {net; $(\ell )$-group; ideal; ideal order; $(D)$-convergence; ideal divergence; net; -group; ideal order; -convergence; ideal divergence},
language = {eng},
number = {4},
pages = {1073-1083},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ideal convergence and divergence of nets in $(\ell )$-groups},
url = {http://eudml.org/doc/246170},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Boccuto, Antonio
AU - Dimitriou, Xenofon
AU - Papanastassiou, Nikolaos
TI - Ideal convergence and divergence of nets in $(\ell )$-groups
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1073
EP - 1083
AB - 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.
LA - eng
KW - net; $(\ell )$-group; ideal; ideal order; $(D)$-convergence; ideal divergence; net; -group; ideal order; -convergence; ideal divergence
UR - http://eudml.org/doc/246170
ER -

References

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