In this paper we introduce the $ℐ$- and ${ℐ}^{*}$-convergence and divergence of nets in $\left(\ell \right)$-groups. We prove some theorems relating different types of convergence/divergence for nets in $\left(\ell \right)$-group setting, in relation with ideals. We consider both order and $\left(D\right)$-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that ${ℐ}^{*}$-convergence/divergence implies $ℐ$-convergence/divergence for every ideal, admissible for...