  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.
