Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1073-1083, 2012

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

Antonio Boccuto, Xenofon Dimitriou, Nikolaos Papanastassiou

A. Boccuto (corresponding author), Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, I-06123 Perugia, Italy, e-mail: boccuto@yahoo.it, boccuto@dmi.unipg.it; X. Dimitriou, Department of Mathematics, University of Athens, Panepistimiopolis, Athens 15784, Greece, e-mail: xenofon11@gmail.com, dxenof@math.uoa.gr; N. Papanastassiou, Department of Mathematics, University of Athens, Panepistimiopolis, Athens 15784, Greece, e-mail: npapanas@math.uoa.gr

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.

Keywords: net, $(\ell)$-group, ideal, ideal order, $(D)$-convergence, ideal divergence

Classification (MSC 2010): 28B15, 54A20


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