The ideal $(a)$ revisited and its applications to the Ellentuck topology

We continue the investigation of the ideal $(a)$ introduced by M. Grande (2001) and its generalizations. We prove that $(a)(\tau_1, \tau_2)$ cannot be represented as a finite intersection of the collections of nowhere dense sets for any topologies stronger than $\tau_1$ and weaker than $\tau_2$. We give a counterexample to the inclusion $(a)(\tau_e, \mathcal B_{\mathrm{EL}})\subseteq (a)(\tau_e, \tau _{\mathrm{EL}})$. We answer a question of Frankowska and Głąb (2019) by constructing a set from $(a)$ which cannot be represented as a finite sum of elements from $(a^\prime )$.