A variety \(\mathcal V\) is minimal if it is equationally complete. A quasivariety is called \(Q\)-universal if for every quasivariety \(K\) of a finite type the lattice \(L_Q (K)\) of all subquasivarieties is a homomorphic image of \(L_Q (Q)\). The author studies varieties \(C_{mn}\) of the so-called Cantor algebras (firstly treated in the early 60s by S. Swierczkowski as algebras having finite bases of different cardinalities). The main results are: (1) \(C_{1n}\) is a minimal variety for each \(n\) (2) \(C_{mn}\) are \(Q\)-universal for all \(1\leq m\leq n\).