FREE OBJECTS IN SOME VARIETIES OF GROUPOIDS - Čupona, Ǵorǵi; Celakoski, Naum (Matematichki Bilten, 1996)
We give a canonical description of free objects in the variety \(V_m,n\) of groupoids which satisfy the law \(x^m y^n=z_1z_2… z_m\), where \(z_i=x\) if \(i\) is odd, \(z_j=y\) if \(j\) is even, and \(m,n\) are integers such that \(m>n≥ 2\). We also consider a derived quasivariety \(V_m,n^D\) of groupoids in which only trivial identities hold.