Ideal convergence and divergence of nets in $(\ell )$-groups

Boccuto, Antonio ; Dimitriou, Xenofon ; Papanastassiou, Nikolaos

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 1073-1083

Abstract (VO)

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.

Détail

DOI : 10.1007/s10587-012-0064-z

Classification : 28B10, 28B15, 54A20
Keywords: net; $(\ell )$-group; ideal; ideal order; $(D)$-convergence; ideal divergence

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