A closure operator respecting the modular j-function

We prove some unconditional cases of the Existential Closedness problem for the modular j-function. For this, we show that for any finitely generated field we can find a “convenient” set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular j-function and its derivatives, one can define a natural closure operator in three equivalent different ways.