Independence relations induced by ideals

Inspired by the non-meager independence $\downarrow^{nm}$ introduced by Krupi\'{n}ski [3], we study further possible independence relations induced by ideals and provide a general framework for this topic. We show that the independence relation $\downarrow^\mathrm{Haar}$ induced by Haar null ideals is a good example for locally compact groups and provide an example showing $\downarrow^\mathrm{Haar}\neq \downarrow^{nm}$. Moreover, we introduce an order on the collection of all well-behaved independence relations and conjecture that $\downarrow^{nm}$ is the maximum one for Polish groups. We prove the conjecture under the extra hypothesis of $\sigma$-compactness. For Lie groups, we discuss $\mathrm{SO}(3)$ and $\mathrm{SE}(2)$ as examples, and prove that the independence relation is unique for nilpotent Lie groups.