# Some further results on ideal summability of nets in ( $$\ell$$ ℓ ) groups

Some further results on ideal summability of nets in ( $$\ell$$ ℓ ) groups 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 $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ order convergence, $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ divergence of nets and its further extensions, namely the notions of $$\mathcal {I}^{\mathcal {K}}$$ I K -order convergence and $$\mathcal {I}^{\mathcal {K}}$$ I K -divergence of nets in a $$(\ell )$$ ( ℓ ) -group and investigate the relation between $$\mathcal {I}$$ I and $$\mathcal {I}^{\mathcal {K}}$$ I K -concepts where a special class of ideals called $$\Lambda P(\mathcal {I},\mathcal {K})-$$ Λ P ( I , K ) - ideals plays very important role. We also introduce, for the first time, the notion of $$\mathcal {I}^{\mathcal {K}}$$ I K -order Cauchy condition and $$\mathcal {I}$$ I -order cluster points of nets in ( $$\ell$$ ℓ ) groups and examine some of its characterizations and its consequences. In particular the role of $$\mathcal {I}$$ I -order cluster points in making the above mentioned Cauchy nets convergent is studied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Some further results on ideal summability of nets in ( $$\ell$$ ℓ ) groups

, Volume 19 (1) – Mar 9, 2014
11 pages

/lp/springer_journal/some-further-results-on-ideal-summability-of-nets-in-ell-groups-KGGXsqtu15
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0282-8
Publisher site
See Article on Publisher Site

### References

• On $$I$$ I -convergence of nets in locally solid Riesz spaces
Das, P; Savas, E

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