On quasi-equational bases for differential groupoids and unary algebras

As is known, there exist $2^\omega$ quasivarieties of differential groupoids and unary algebras with no independent quasi-equational basis. In the present article, we show that there exist $2^\omega$ such quasivarieties with an $\omega$-independent quasi-equational basis. We also find a recursive independent quasi-equational basis for the intersection of those quasivarieties.