Some further results on ideal summability of nets in (Formula Presented.) groups

Das P., Savaş E.

Positivity, vol.19, no.1, pp.53-63, 2015 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 1
  • Publication Date: 2015
  • Doi Number: 10.1007/s11117-014-0282-8
  • Journal Name: Positivity
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.53-63
  • Keywords: (ℓ)-group, Filter, I-divergence, I-order-cluster point, I-order-convergence/Cauchy condition, IK-divergence, IK-order-convergence/Cauchy condition, Ideal, Net
  • Uşak University Affiliated: No

Abstract

In this paper we continue in the line of recent investigation of order summability of nets using ideals by Boccuto et al. (Czechoslovak Math. J. 62(137):1073–1083 2012; J. Appl. Anal. 20(1), 2014) where they had introduced the notions of I and (Formula Presented.) order convergence, I and (Formula Presented.) divergence of nets and its further extensions, namely the notions of (Formula Presented.)-order convergence and (Formula Presented.)-divergence of nets in a (Formula Presented.)-group and investigate the relation between (Formula Presented.)-concepts where a special class of ideals called (Formula Presented.)-ideals plays very important role. We also introduce, for the first time, the notion of (Formula Presented.)-order Cauchy condition and (Formula Presented.)-order cluster points of nets in ((Formula Presented.)) groups and examine some of its characterizations and its consequences. In particular the role of (Formula Presented.)-order cluster points in making the above mentioned Cauchy nets convergent is studied.