FREE OBJECTS IN SOME VARIETIES OF GROUPOIDS

Abstract

We give a canonical description of free objects in the variety \(\mathcal{V}_{m,n}\) of groupoids which satisfy the law \(x^m y^n=z_1z_2\cdots z_m\), where \(z_i=x\) if \(i\) is odd, \(z_j=y\) if \(j\) is even, and \(m,n\) are integers such that \(m>n\geq 2\). We also consider a derived quasivariety \(\mathcal{V}_{m,n}^{D}\) of groupoids in which only trivial identities hold.