Generalized independence

We explore different generalizations of the classical concept of independent families on ω following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under ${\left(D\ell \right)}_{\kappa }^{⁎}$ we can get strongly κ-independent families of size $2^{\kappa }$ and present an equivalence of $\mathrm{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $C$-independent families, where $C$ is the club filter. Also we show a relationship between the existence of $J$-independent families and the saturation of the ideal $J$.