Quasigroup covers of division groupoids

Let G be a division groupoid that is not a quasigroup. For each regular cardinal alpha > vertical bar G vertical bar we construct a quasigroup Q on G x alpha that is a quasigroup cover of G (i.e., G is a homomorphic image of Q and G is not an image of any quasigroup that is a proper factor of Q). We also show how to easily obtain quasigroup covers from free quasigroups.