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

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 $\mathcal {I}$ and $\mathcal {I}^*$ order convergence, $\mathcal {I}$ and $\mathcal {I}^*$ divergence of nets and its further extensions, namely the notions of $\mathcal {I}^{\mathcal {K}}$-order convergence and $\mathcal {I}^{\mathcal {K}}$-divergence of nets in a $(\ell )$-group and investigate the relation between $\mathcal {I}$ and $\mathcal {I}^{\mathcal {K}}$-concepts where a special class of ideals called $\Lambda P(\mathcal {I},\mathcal {K})-$ideals plays very important role. We also introduce, for the first time, the notion of $\mathcal {I}^{\mathcal {K}}$-order Cauchy condition and $\mathcal {I}$-order cluster points of nets in ($\ell$) groups and examine some of its characterizations and its consequences. In particular the role of $\mathcal {I}$-order cluster points in making the above mentioned Cauchy nets convergent is studied.